Flexible Inner-Outer Krylov Subspace Methods

Flexible Krylov methods refers to a class of methods which accept preconditioning that can change from one step to the next. Given a Krylov subspace method, such as CG, GMRES, QMR, etc. for the solution of a linear system Ax=b, instead of having a fixed preconditioner M and the (right) preconditioned equation AM-1 y = b (Mx =y), one may have a different matrix, say Mk, at each step. In this paper, the case where the preconditioner itself is a Krylov subspace method is studied. There are several papers in the literature where such a situation is presented and numerical examples given. A general theory is provided encompassing many of these cases, including truncated methods. The overall space where the solution is approximated is no longer a Krylov subspace but a subspace of a larger Krylov space. We show how this subspace keeps growing as the outer iteration progresses, thus providing a convergence theory for these inner-outer methods. Numerical tests illustrate some important implementation aspects that make the discussed inner-outer methods very appealing in practical circumstances.

[1]  Peter N. Brown,et al.  A Theoretical Comparison of the Arnoldi and GMRES Algorithms , 1991, SIAM J. Sci. Comput..

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

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

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

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

[6]  Oliver G. Ernst,et al.  Analysis of acceleration strategies for restarted minimal residual methods , 2000 .

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

[8]  Yousef Saad,et al.  Deflated and Augmented Krylov Subspace Techniques , 1997, Numer. Linear Algebra Appl..

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

[10]  E. Sturler,et al.  Nested Krylov methods based on GCR , 1996 .

[11]  G. Stewart.  Backward error bounds for approximate Krylov subspaces , 2002 .

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

[13]  Michele Benzi,et al.  Preconditioning a mixed discontinuous finite element method for radiation diffusion , 2004, Numer. Linear Algebra Appl..

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

[15]  K. Burrage,et al.  Restarted GMRES preconditioned by deflation , 1996 .

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

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

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

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

[20]  C. Vuik.  New insights in GMRES-like methods with variable preconditioners , 1995 .

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

[22]  Courtenay T. Vaughan,et al.  Efficient Polynomial Preconditioning for the Conjugate Gradient Method , 1990, SIAM J. Sci. Comput..

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

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

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

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

[27]  Kesheng Wu,et al.  DQGMRES: a Direct Quasi-minimal Residual Algorithm Based on Incomplete Orthogonalization , 1996, Numer. Linear Algebra Appl..

[28]  G. Stewart,et al.  Matrix Perturbation Theory , 1990 .

[29]  J. Meijerink,et al.  An iterative solution method for linear systems of which the coefficient matrix is a symmetric -matrix , 1977 .

[30]  Homer F. Walker,et al.  Residual Smoothing Techniques for Iterative Methods , 1994, SIAM J. Sci. Comput..

[31]  Wayne Joubert,et al.  A Robust GMRES-Based Adaptive Polynomial Preconditioning Algorithm for Nonsymmetric Linear Systems , 1994, SIAM J. Sci. Comput..

[32]  Daniel B. Szyld,et al.  FQMR: A Flexible Quasi-Minimal Residual Method with Inexact Preconditioning , 2001, SIAM J. Sci. Comput..

[33]  Thomas A. Manteuffel,et al.  Minimal Residual Method Stronger than Polynomial Preconditioning , 1996, SIAM J. Matrix Anal. Appl..

[34]  David G. Luenberger,et al.  Linear and Nonlinear Programming: Second Edition , 2003 .

[35]  Valeria Simoncini,et al.  On the Convergence of Restarted Krylov Subspace Methods , 2000, SIAM J. Matrix Anal. Appl..

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