A Flexible Generalized Conjugate Residual Method with Inner Orthogonalization and Deflated Restarting

  • Authors:

  • L. M. Carvalho;S. Gratton;R. Lago;X. Vasseur

  • Affiliations:

  • luizmc@gmail.com;serge.gratton@enseeiht.fr;lago@cerfacs.fr;vasseur@cerfacs.fr

  • Venue:
  • SIAM Journal on Matrix Analysis and Applications
  • Year:
  • 2011

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Abstract

This work is concerned with the development and study of a minimum residual norm subspace method based on the generalized conjugate residual method with inner orthogonalization (GCRO) method that allows flexible preconditioning and deflated restarting for the solution of nonsymmetric or non-Hermitian linear systems. First we recall the main features of flexible generalized minimum residual with deflated restarting (FGMRES-DR), a recently proposed algorithm of the same family but based on the GMRES method. Next we introduce the new inner-outer subspace method named FGCRO-DR. A theoretical comparison of both algorithms is then made in the case of flexible preconditioning. It is proved that FGCRO-DR and FGMRES-DR are algebraically equivalent if a collinearity condition is satisfied. While being nearly as expensive as FGMRES-DR in terms of computational operations per cycle, FGCRO-DR offers the additional advantage to be suitable for the solution of sequences of slowly changing linear systems (where both the matrix and right-hand side can change) through subspace recycling. Numerical experiments on the solution of multidimensional elliptic partial differential equations show the efficiency of FGCRO-DR when solving sequences of linear systems.