Restarted Hessenberg method for solving shifted nonsymmetric linear systems

It is known that the restarted full orthogonalization method (FOM) outperforms the restarted generalized minimum residual (GMRES) method in several circumstances for solving shifted linear systems when the shifts are handled simultaneously. Many variants of them have been proposed to enhance their performance. We show that another restarted method, the restarted Hessenberg method [M. Heyouni, Methode de Hessenberg Generalisee et Applications, Ph.D. Thesis, Universite des Sciences et Technologies de Lille, France, 1996] based on Hessenberg procedure, can effectively be employed, which can provide accelerating convergence rate with respect to the number of restarts. Theoretical analysis shows that the new residual of shifted restarted Hessenberg method is still collinear with each other. In these cases where the proposed algorithm needs less enough CPU time elapsed to converge than the earlier established restarted shifted FOM, weighted restarted shifted FOM, and some other popular shifted iterative solvers based on the short-term vector recurrence, as shown via extensive numerical experiments involving the recent popular applications of handling the time fractional differential equations.

[1]  Jian-Jun Zhang,et al.  Restarted Gmres Augmented With Eigenvectors For Shifted Linear Systems * Supported by the National Natural Science Foundation of China and the Science and Technology Developing Foundation of University in Shanghai of China , 2003, Int. J. Comput. Math..

[2]  Gui-Ding Gu,et al.  Restarted GMRES augmented with harmonic Ritz vectors for shifted linear systems , 2005, Int. J. Comput. Math..

[3]  Gang Wu,et al.  A Preconditioned and Shifted GMRES Algorithm for the PageRank Problem with Multiple Damping Factors , 2012, SIAM J. Sci. Comput..

[4]  B. Jegerlehner.  Krylov space solvers for shifted linear systems , 1996, hep-lat/9612014.

[5]  Tomohiro Sogabe,et al.  ON A WEIGHTED QUASI-RESIDUAL MINIMIZATION STRATEGY FOR SOLVING COMPLEX SYMMETRIC SHIFTED LINEAR SYSTEMS , 2008 .

[6]  T. Sogabe,et al.  A Numerical Method for Calculating the Green's Function Arising from Electronic Structure Theory , 2007 .

[7]  Lei Du,et al.  IDR(s) for solving shifted nonsymmetric linear systems , 2015, J. Comput. Appl. Math..

[8]  Zhishun A. Liu,et al.  A Look Ahead Lanczos Algorithm for Unsymmetric Matrices , 1985 .

[9]  Jun-Feng Yin and Guo-Jian Yin.  Restarted Full Orthogonalization Method with Deflation for Shifted Linear Systems , 2014 .

[10]  Gerard L. G. Sleijpen,et al.  Flexible and multi-shift induced dimension reduction algorithms for solving large sparse linear systems , 2011 .

[11]  Gerard L. G. Sleijpen,et al.  Flexible and multi-shift induced dimension reduction algorithms for solving large sparse linear systems , 2015, Numer. Linear Algebra Appl..

[12]  Roberto Garrappa,et al.  On the use of matrix functions for fractional partial differential equations , 2011, Math. Comput. Simul..

[13]  Andreas Frommer,et al.  BiCGStab(ℓ) for Families of Shifted Linear Systems , 2003, Computing.

[14]  Ting-Zhu Huang,et al.  Variants of the CMRH method for solving multi-shifted non-Hermitian linear systems , 2016 .

[15]  Hassane Sadok,et al.  Algorithms for the CMRH method for dense linear systems , 2015, Numerical Algorithms.

[16]  Roberto Garrappa,et al.  A family of Adams exponential integrators for fractional linear systems , 2013, Comput. Math. Appl..

[17]  R. Morgan,et al.  Deflated GMRES for systems with multiple shifts and multiple right-hand sides☆ , 2007, 0707.0502.

[18]  Martin B. van Gijzen,et al.  Preconditioned Multishift BiCG for ℋ2-Optimal Model Reduction , 2017, SIAM J. Matrix Anal. Appl..

[19]  Lei,et al.  A FLEXIBLE PRECONDITIONED ARNOLDI METHOD FOR SHIFTED LINEAR SYSTEMS , 2007 .

[20]  R. Takayama,et al.  Linear algebraic calculation of the Green’s function for large-scale electronic structure theory , 2006 .

[21]  I. Podlubny.  Fractional differential equations , 1998 .

[22]  J. H. Wilkinson.  The algebraic eigenvalue problem , 1966 .

[23]  Ting-Zhu Huang,et al.  BiCR-type methods for families of shifted linear systems , 2014, Comput. Math. Appl..

[24]  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..

[25]  Chuanqing Gu,et al.  A flexible CMRH algorithm for nonsymmetric linear systems , 2014 .

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

[27]  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..

[28]  Peter K. Kitanidis,et al.  A Flexible Krylov Solver for Shifted Systems with Application to Oscillatory Hydraulic Tomography , 2012, SIAM J. Sci. Comput..

[29]  Andreas Frommer,et al.  MANY MASSES ON ONE STROKE: ECONOMIC COMPUTATION OF QUARK PROPAGATORS , 1995 .

[30]  Sheehan Olver,et al.  Shifted GMRES for oscillatory integrals , 2010, Numerische Mathematik.

[31]  Karl Meerbergen,et al.  The Solution of Parametrized Symmetric Linear Systems , 2002, SIAM J. Matrix Anal. Appl..

[32]  V. Simoncini,et al.  Iterative system solvers for the frequency analysis of linear mechanical systems , 2000 .

[33]  Gang Wu,et al.  Preconditioning the Restarted and Shifted Block FOM Algorithm for Matrix Exponential Computation , 2014, 1405.0707.

[34]  Y. Saad,et al.  Arnoldi methods for large Sylvester-like observer matrix equations, and an associated algorithm for partial spectrum assignment , 1991 .

[35]  L. Trefethen,et al.  Talbot quadratures and rational approximations , 2006 .

[36]  Michael K. Ng,et al.  Galerkin Projection Methods for Solving Multiple Linear Systems , 1999, SIAM J. Sci. Comput..

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

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

[39]  Bruno Lang,et al.  An iterative method to compute the sign function of a non-Hermitian matrix and its application to the overlap Dirac operator at nonzero chemical potential , 2007, Comput. Phys. Commun..

[40]  H. V. D. Vorst,et al.  The superlinear convergence behaviour of GMRES , 1993 .

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

[42]  R. Freund.  Solution of shifted linear systems by quasi-minimal residual iterations , 1993 .

[43]  Hassane Sadok,et al.  A new implementation of the CMRH method for solving dense linear systems , 2008 .

[44]  Ting-Zhu Huang,et al.  Computers and Mathematics with Applications Restarted Weighted Full Orthogonalization Method for Shifted Linear Systems , 2022 .

[45]  V. Simoncini.  Restarted Full Orthogonalization Method for Shifted Linear Systems , 2003 .

[46]  Mohammed Heyouni.  Méthode de Hessenberg généralisée et applications , 1996 .

[47]  Andreas Frommer,et al.  Restarted GMRES for Shifted Linear Systems , 1998, SIAM J. Sci. Comput..

[48]  I. Moret,et al.  An interpolatory approximation of the matrix exponential based on Faber polynomials , 2001 .

[49]  Lloyd N. Trefethen,et al.  Parabolic and hyperbolic contours for computing the Bromwich integral , 2007, Math. Comput..

[50]  Gérard Meurant,et al.  On the convergence of Q-OR and Q-MR Krylov methods for solving nonsymmetric linear systems , 2016 .

[51]  Shuying Zhai,et al.  A novel high-order ADI method for 3D fractionalconvection–diffusion equations ☆ ☆☆ , 2015 .

[52]  Martin B. van Gijzen,et al.  Nested Krylov Methods for Shifted Linear Systems , 2014, SIAM J. Sci. Comput..

[53]  Mei Han An,et al.  accuracy and stability of numerical algorithms , 1991 .

[54]  Hassane Sadok,et al.  On a variable smoothing procedure for Krylov subspace methods , 1998 .

[55]  Hassane Sadok,et al.  A new look at CMRH and its relation to GMRES , 2012 .

[56]  Tomohiro Sogabe,et al.  An Extension of the COCR Method to Solving Shifted Linear Systems with Complex Symmetric Matrices , 2011 .

[57]  Eric de Sturler,et al.  Recycling BiCG with an Application to Model Reduction , 2010, SIAM J. Sci. Comput..

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

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

[60]  Ronald B. Morgan,et al.  A Restarted GMRES Method Augmented with Eigenvectors , 1995, SIAM J. Matrix Anal. Appl..

[61]  Peter K. Kitanidis,et al.  A fast algorithm for parabolic PDE-based inverse problems based on Laplace transforms and flexible Krylov solvers , 2014, J. Comput. Phys..

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

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

[64]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[65]  Wei Wang,et al.  A Fourth-Order Compact BVM Scheme for the Two-Dimensional Heat Equations , 2008, CSC.

[66]  Yong Zhang,et al.  Efficient preconditioner updates for unsymmetric shifted linear systems , 2014, Comput. Math. Appl..

[67]  Gene H. Golub,et al.  Matrix computations , 1983 .

[68]  Mehdi Dehghan,et al.  Generalized product-type methods based on bi-conjugate gradient (GPBiCG) for solving shifted linear systems , 2017 .

[69]  知広 曽我部.  Extensions of the conjugate residual method , 2006 .