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

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 non-symmetric 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.
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Luiz Mariano Carvalho, Serge Gratton, Rafael Lago, Xavier Vasseur. A Flexible Generalized Conjugate Residual Method with Inner Orthogonalization and Deflated Restarting. SIAM Journal on Matrix Analysis and Applications, Society for Industrial and Applied Mathematics, 2011, 32 (4), pp.1212 - 1235. ⟨10.1137/100786253⟩. ⟨hal-00650239v2⟩



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