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Augmented GMRES-type methods. (English) Zbl 1199.65096
The paper addresses the GMRES method and a slightly modified method, RRGMRES, and investigates some options to extend the involved Krylov subspaces with spaces generated by a small number of vectors. These so-called augmented Krylov subspaces are not defined with the help of additional approximate eigenvectors or Ritz-vectors, as has been done before, but instead they use spaces that enable the representation of certain known non-smooth features of the desired solution such as jumps. After a description of the implementation of augmented GMRES or RRGMRES and some general theoretical results on these augmented methods, numerical experiments with well-posed and ill-posed problems are presented. They show that the proposed augmented methods can yield satisfactory approximations of the solution in a lower number of iterations than standard GMRES or RRGMRES.

65F10 Iterative numerical methods for linear systems
65F22 Ill-posedness and regularization problems in numerical linear algebra
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