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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.

MSC:
65F10 Iterative numerical methods for linear systems
65F22 Ill-posedness and regularization problems in numerical linear algebra
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References:
[1] Iterative Methods for Sparse Linear Systems (2nd edn). SIAM: Philadelphia, 2003.
[2] Saad, SIAM Journal on Scientific and Statistical Computing 7 pp 856– (1986)
[3] Calvetti, Linear Algebra and its Applications 316 pp 157– (2000)
[4] Calvetti, International Journal of Applied Mathematics and Computer Science 11 pp 1069– (2001)
[5] Reichel, SIAM Journal on Matrix Analysis and Applications 26 pp 1001– (2005)
[6] Chapman, Numerical Linear Algebra with Applications 4 pp 43– (1997)
[7] Morgan, SIAM Journal on Matrix Analysis and Applications 16 pp 1154– (1995)
[8] Morgan, SIAM Journal on Matrix Analysis and Applications 21 pp 1112– (2000)
[9] Morgan, SIAM Journal on Scientific Computing 24 pp 20– (2002)
[10] Saad, SIAM Journal on Matrix Analysis and Applications 18 pp 435– (1997)
[11] Calvetti, Linear Algebra and its Applications 362 pp 257– (2003) · Zbl 1069.74011
[12] Baglama, Journal of Computational and Applied Mathematics 198 pp 332– (2007)
[13] Rank Deficient and Discrete Ill-Posed Problems. SIAM: Philadelphia, 1998. · Zbl 0890.65037
[14] Calvetti, Numerische Mathematik 91 pp 605– (2002)
[15] Calvetti, BIT 42 pp 44– (2002)
[16] Conjugate Gradient Type Methods for Ill-Posed Problems. Longman: Harlow, 1995.
[17] Hanke, Surveys on Mathematics for Industry 3 pp 253– (1993)
[18] Hansen, Numerical Linear Algebra with Applications 3 pp 513– (1996)
[19] Baart, IMA Journal of Numerical Analysis 2 pp 241– (1982)
[20] Hansen, Numerical Algorithms 6 pp 1– (1994)
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