FQMR: A Flexible Quasi-Minimal Residual Method with Inexact Preconditioning

A flexible version of the QMR algorithm is presented which allows for the use of a different preconditioner at each step of the algorithm. In particular, inexact solutions of the preconditioned equations are allowed, as well as the use of an (inner) iterative method as a preconditioner. Several theorems are presented relating the norm of the residual of the new method with the norm of the residual of other methods, including QMR and flexible GMRES (FGMRES). In addition, numerical experiments are presented which illustrate the convergence of flexible QMR (FQMR), and show that in certain cases FQMR can produce approximations with lower residual norms than QMR.

[1]  荻野 哲男.  Indiana University 留学記 , 2001 .

[2]  Yousef Saad,et al.  Rational Approximation Preconditioners for General Sparse Linear Systems , 1999 .

[3]  D. Szyld,et al.  H-Splittings and two-stage iterative methods , 1992 .

[4]  E. Sturler,et al.  Truncation Strategies for Optimal Krylov Subspace Methods , 1999 .

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

[6]  Roland W. Freund,et al.  An Implementation of the QMR Method Based on Coupled Two-Term Recurrences , 1994, SIAM J. Sci. Comput..

[7]  Jesús M. Castel,et al.  On parallel two-stage methods for Hermitian positive definite matrices with applications to preconditioning. , 2001 .

[8]  孝蔵 藤井,et al.  NASA Ames Research Centerにおける数値流体力学研究 , 1985 .

[9]  Yvan Notay.  Flexible Conjugate Gradients , 2000, SIAM J. Sci. Comput..

[10]  V. Frayssé,et al.  A relaxation strategy for inexact matrix-vector products for Krylov methods , 2000 .

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

[12]  N. Nachtigal.  A look-ahead variant of the Lanczos algorithm and its application to the quasi-minimal residual method for non-Hermitian linear systems. Ph.D. Thesis - Massachusetts Inst. of Technology, Aug. 1991 , 1991 .

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

[14]  Anne Greenbaum,et al.  Iterative methods for solving linear systems , 1997, Frontiers in applied mathematics.

[15]  Gene H. Golub,et al.  Inexact Preconditioned Conjugate Gradient Method with Inner-Outer Iteration , 1999, SIAM J. Sci. Comput..

[16]  Cornelis Vuik,et al.  GMRESR: a family of nested GMRES methods , 1994, Numer. Linear Algebra Appl..

[17]  Mark T. Jones,et al.  Two-Stage and Multisplitting Methods for the Parallel Solution of Linear Systems , 1992, SIAM J. Matrix Anal. Appl..

[18]  R. Dembo,et al.  INEXACT NEWTON METHODS , 1982 .

[19]  Dianne P. O'Leary,et al.  A Multigrid Method Enhanced by Krylov Subspace Iteration for Discrete Helmholtz Equations , 2001, SIAM J. Sci. Comput..

[20]  Gene H. Golub,et al.  Adaptively Preconditioned GMRES Algorithms , 1998, SIAM J. Sci. Comput..

[21]  M. Eiermann,et al.  Geometric aspects of the theory of Krylov subspace methods , 2001, Acta Numerica.

[22]  Yousef Saad,et al.  A Flexible Inner-Outer Preconditioned GMRES Algorithm , 1993, SIAM J. Sci. Comput..

[23]  Iain S. Duff,et al.  Sparse matrix test problems , 1982 .

[24]  G. Golub,et al.  The convergence of inexact Chebyshev and Richardson iterative methods for solving linear systems , 1988 .

[25]  O. Axelsson,et al.  A black box generalized conjugate gradient solver with inner iterations and variable-step preconditioning , 1991 .