Chebyshev acceleration of iterative refinement

It is well known that the FGMRES algorithm can be used as an alternative to iterative refinement and, in some instances, is successful in computing a backward stable solution when iterative refinement fails to converge. In this study, we analyse how variants of the Chebyshev algorithm can also be used to accelerate iterative refinement without loss of numerical stability and at a computational cost at each iteration that is less than that of FGMRES and only marginally greater than that of iterative refinement. A component-wise error analysis of the procedure is presented and numerical tests on selected sparse test problems are used to corroborate the theory.

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

[2]  Zdenek Strakos,et al.  Accuracy of Two Three-term and Three Two-term Recurrences for Krylov Space Solvers , 2000, SIAM J. Matrix Anal. Appl..

[3]  Jennifer A. Scott,et al.  HSL_MA97 : a bit-compatible multifrontal code for sparse symmetric systems , 2011 .

[4]  T. Manteuffel.  Adaptive procedure for estimating parameters for the nonsymmetric Tchebychev iteration , 1978 .

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

[6]  Nicholas J. Higham,et al.  Accuracy and stability of numerical algorithms, Second Edition , 2002 .

[7]  RobertF Brown,et al.  A topological introduction to nonlinear analysis , 1993 .

[8]  I. Duff,et al.  Using FGMRES to obtain backward stability in mixed precision , 2008 .

[9]  H. Rutishauser.  Theory of Gradient Methods , 1959 .

[10]  J. Demmel,et al.  Solving Sparse Linear Systems with Sparse Backward Error , 2015 .

[11]  Nicholas J. Higham,et al.  INVERSE PROBLEMS NEWSLETTER , 1991 .

[12]  R. Skeel.  Iterative refinement implies numerical stability for Gaussian elimination , 1980 .

[13]  David M. Young,et al.  Applied Iterative Methods , 2004 .

[14]  Martin H. Gutknecht,et al.  The Chebyshev iteration revisited , 2002, Parallel Comput..

[15]  T. Manteuffel.  The Tchebychev iteration for nonsymmetric linear systems , 1977 .

[16]  Jennifer A. Scott,et al.  A fast and robust mixed-precision solver for the solution of sparse symmetric linear systems , 2010, TOMS.

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

[18]  Jennifer A. Scott,et al.  A note on the solve phase of a multicore solver , 2010 .

[19]  Serge Gratton,et al.  A Note on GMRES Preconditioned by a Perturbed L D LT Decomposition with Static Pivoting , 2007, SIAM J. Sci. Comput..