Journal of Mathematical Sciences

, Volume 224, Issue 6, pp 890–899 | Cite as

Iterative Processes in the Krylov–Sonneveld Subspaces


The paper presents a generalized block version of the Induced Dimension Reduction (IDR) methods in comparison with the Multi–Preconditioned Semi-Conjugate Direction (MPSCD) algorithms in Krylov subspaces with deflation and low-rank matrix approximation. General and individual orthogonality and variational properties of these two methodologies are analyzed. It is demonstrated, in particular, that for any sequence of Krylov subspaces with increasing dimensions there exists a sequence of the corresponding shrinking subspaces with decreasing dimensions. The main conclusion is that the IDR procedures, proposed by P. Sonneveld and other authors, are not an alternative to but a further development of the general principles of iterative processes in Krylov subspaces.


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  1. 1.
    P. Wesseling and P. Sonneveld, “Numerical experiments with a multiple grid and a preconditioned Lanczos type method,” Lect. Notes Math., 771, 543–562 (1980).MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    P. Sonneveld and M. B. Van Gijzen, “IDR(s): a family of simple and fast algorithms for solving large nonsymmetric systems of linear equations,” SIAM J. Sci. Comput., 31, No. 2, 1035–1062 (2008/09).MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    M. H. Gutknecht, “IDR explained,” Electron. Trans. Numer. Anal., 36, 126–148 (2009/10).MathSciNetMATHGoogle Scholar
  4. 4.
    Y. Onoue, S. Fujino, and N. Nakashima, “An overview of a family of new iterative methods based on IDR theorem and its estimation,” in: Proceedings of the International Multi-Conference of Engineers and Computer Scientists, 2009, Vol. 11, IMECS, Hong Kong (2009), pp. 2129–2134.Google Scholar
  5. 5.
    V. Simoncini and D. B. Szyld, “Interpreting IDR as a Petrov–Galerkin method,” SIAM J. Sci. Comput., 32, No. 4, 1898–1912 (2010).MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    M. Tanio and M. Sugihara, “GBi-CGSTAB(s,l): IDR(s) with higher-order stabilization polynomials,” J. Comput. Appl. Math., 235, No. 3, 765–784 (2010).MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    G. L. Sleijpen, P. Sonneveld, and M. B. Van Gijzen, “Bi-CGStab as an induced dimension reduction method,” Appl. Numer. Math., 60, No. 11, 1100–1114 (2010).MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    G. L. Sleijpen and M. B. Van Gijzen, “Exploiting BiCGStab(l) strategies to induce dimension reduction,” SIAM J. Sci. Comput., 32 (5), 2687–2709 (2010).MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    P. Sonneveld, “On the convergence behavior of IDR(s) and related methods,” SIAM J. Sci. Comput., 34 (5), 2576–2598 (2012).MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Y. Saad, Iterative Methods for Sparse Linear Systems (Second edition), Society for Industrial and Applied Mathematics (2003).Google Scholar
  11. 11.
    V. P. Il’in, Methods and Technologies of Finite Elements [in Russian], Izd. IVMiMG SO RAN, Novosibirsk (2007).Google Scholar
  12. 12.
    G. L. Sleijpen and D. R. Fokkema, “BiCGStab(l) for linear equations involving unsymmetric matrices with complex spectrum,” Electron. Trans. Numer. Anal., 1, 11–32 (1993).MathSciNetMATHGoogle Scholar
  13. 13.
    V. P. Il’in, “Biconjugate direction methods in Krylov subspaces,” Sib. Zh. Ind. Mat., 11, No. 4, 47–60 (2008).MATHGoogle Scholar
  14. 14.
    M. B. Van Gijzen, G. I. Sleijpen, and J. -P. Zemke, “Flexible and multi-shift induced dimension reduction algorithms for solving large sparse linear systems,” Numer. Linear Algebra Appl., 22, 1–25 (2014).MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    M. B. Van Gijzen and P. Sonneveld, “Algorithm 913: an elegant IDR(s) variant that efficiently exploits biorthogonality properties,” ACM Trans. Math. Software, 38, No. 1, 5.1–5.19 (2011).MathSciNetMATHGoogle Scholar
  16. 16.
    O. Rendel, A. Rizvanolli, and J.-P. Zemke, “IDR: A new generation of Krylov subspace methods,” Linear Algebra Appl., 439, No. 4, 1040–1061 (2013).MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    R. Astudillo and M. B. Van Gijzen, “A restarted induced dimension reduction method to approximate eigenpairs of large unsymmetric matrices,” J. Comput. Appl. Math., 296, 24–35 (2016).MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    V. P. Il’in, “Methods of semiconjugate directions,” Russ. J. Numer. Anal. Math. Model., 23, No. 4, 369–387 (2008).MathSciNetMATHGoogle Scholar
  19. 19.
    V. P. Il’in and E. A. Itskovich, “On the semi-conjugate direction methods with dynamic preconditioning,” J. Appl. Ind. Math., 3, No. 2, 222–233 (2009).MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    S. C. Eisenstat, H. C. Elman, and M. H. Schultz, “Variational iterative methods for nonsymmetric systems of linear equations,” SIAM J. Numer. Anal., 20, No. 3, 345–357 (1983).MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    J. Y. Yuan, G. H. Golub, R. J. Plemmons, and W. A. Cecilio, “Semi-conjugate direction methods for real positive definite systems,” BIT, 44, No. 1, 189–207 (2004).MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    R. Nicolaides, “Deflation of conjugate gradients with applications to boundary value problems,” SIAM J. Numer. Anal., 24, 355–365 (1987).MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    L. Yu. Kolotilina, “Twofold deflation preconditioning of linear algebraic systems. I. Theory,” Zap. Nauchn. Semin. POMI, 229, 95–152 (1995).MATHGoogle Scholar
  24. 24.
    O. Coulaud, L. Giraud, P. Ramet, and X. Vasseur, Deflation and augmentation techniques in Krylov subspace methods for the solution of linear systems, INRIA, Bordeaux-Sud Ouest, France, Res. Rep. No. 8265 (2013).Google Scholar
  25. 25.
    M. H. Gutknecht, “Deflated and augmented Krylov subspace methods: A framework for deflated BICG and related solvers,” SIAM J. Matrix Anal. Appl., 35, No. 4, 1444–1466 (2014).MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Y. L. Gurieva and V. P. Il’in, “Parallel approaches and technologies of domain decomposition methods,” Zap. Nauchn. Semin. POMI, 428, 89–106 (2014).MATHGoogle Scholar
  27. 27.
    V. P. Il’in, “Problems of parallel solution of large systems of linear algebraic equations,” Zap. Nauchn. Semin. POMI, 439, 112–127 (2015).Google Scholar
  28. 28.
    R. Bridson and C. Greif, “A multipreconditioned conjugate gradient algorithm,” SIAM J. Matrix Anal. Appl., 27, No. 4, 1056–1068 (2006).MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    C. L. Lawson and R. J. Hanson, Solving Least Squares Problems [Russian translation], Nauka, Moscow (1986).Google Scholar

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© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Institute of Computational Mathematics and Mathematical GeophysicsSB RAS and Novosibirsk State UniversityNovosibirskRussia

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