BIT Numerical Mathematics

, Volume 34, Issue 2, pp 177–204 | Cite as

A numerical study of optimized sparse preconditioners

  • A. M. Bruaset
  • A. Tveito


Preconditioning strategies based on incomplete factorizations and polynomial approximations are studied through extensive numerical experiments. We are concerned with the question of the optimal rate of convergence that can be achieved for these classes of preconditioners.

Our conclusion is that the well-known Modified Incomplete Cholesky factorization (MIC), cf. e.g., Gustafsson [20], and the polynomial preconditioning based on the Chebyshev polynomials, cf. Johnson, Micchelli and Paul [22], have optimal order of convergence as applied to matrix systems derived by discretization of the Poisson equation. Thus for the discrete two-dimensional Poisson equation withn unknowns,O(n1/4) andO(n1/2) seem to be the optimal rates of convergence for the Conjugate Gradient (CG) method using incomplete factorizations and polynomial preconditioners, respectively. The results obtained for polynomial preconditioners are in agreement with the basic theory of CG, which implies that such preconditioners can not lead to improvement of the asymptotic convergence rate.

By optimizing the preconditioners with respect to certain criteria, we observe a reduction of the number of CG iterations, but the rates of convergence remain unchanged.

AMS subject classification

65F10 15A06 65F90 65K10 

Key words

Conjugate gradient method preconditioning incomplete factorization polynomial preconditioner matrix-free method Fourier analysis 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    E. Arge, M. Dæhlen, and A. Tveito,Box spline interpolation; a computational study J. Comput. Appl. Math., 44 (1992), pp. 303–329.Google Scholar
  2. 2.
    S. F. Ashby,Polynomial preconditioning for conjugate gradient methods, Department of Computer Science, University of Illinois at Urbana-Champaign, Illinois, Ph.D. thesis, 1987. (Report No. UIUCDCS-R-87-1355.)Google Scholar
  3. 3.
    S. F. Ashby,Minimax polynomial preconditioning for Hermitian linear systems SIAM J. Matrix Anal., 12 (1991), pp. 766–789.Google Scholar
  4. 4.
    S. F. Ashby, M. J. Holst, T. A. Manteuffel, and P. E. Saylor,The role of the inner product in stopping criteria for conjugate gradient iterations, Report UCRL-JC-112586, Comp. & Math. Research Division, Lawrence Livermore National Lab., 1992.Google Scholar
  5. 5.
    S. F. Ashby, T. A. Manteuffel, and J. S. Otto,A comparison of adaptive Chebyshev and least squares polynomial preconditioning for Hermitian positive definite linear systems SIAM J. Sci. Stat. Comput., 13 (1992), pp. 1–29.Google Scholar
  6. 6.
    S. F. Ashby, T. A. Manteuffel, and P. E. Saylor,Adaptive polynomial preconditioning for Hermitian linear systems BIT, 29 (1989), pp. 583–609.Google Scholar
  7. 7.
    S. F. Ashby, T. A. Manteuffel, and P. E. Saylor,A taxonomy for conjugate gradient methods SIAM J. Numer. Anal., 27 (1990), pp. 1542–1568.Google Scholar
  8. 8.
    O. Axelsson and G. Lindskog,On the eigenvalue distribution of a class of preconditioning methods Numer. Math., 48 (1986), pp. 479–498.Google Scholar
  9. 9.
    O. Axelsson and G. Lindskog,On the rate of convergence of the preconditioned conjugate gradient method Numer. Math., 48 (1986), pp. 499–523.Google Scholar
  10. 10.
    P. N. Brown and A. C. Hindmarsh,Matrix-free methods for stiff systems of ODE's SIAM J. Numer. Anal., 23 (1986), pp. 610–638.Google Scholar
  11. 11.
    T. F. Chan,Fourier analysis of relaxed incomplete factorization preconditioners SIAM J. Sci. Stat. Comput., 12 (1991), pp. 668–680.Google Scholar
  12. 12.
    T. F. Chan and H. C. Elman,Fourier analysis of iterative methods for elliptic problems SIAM Review, 31 (1989), pp. 20–49.Google Scholar
  13. 13.
    P. Concus, G. H. Golub, and D. O'Leary,A generalized conjugate gradient method for the numerical solution of elliptic partial differential equations, in Sparse Matrix Computations, J. R. Bunch and D. J. Rose, eds., Academic Press, 1976, pp. 309–332.Google Scholar
  14. 14.
    S. D. Conte and C. de Boor,Elementary Numerical Analysis, McGraw-Hill, 1981.Google Scholar
  15. 15.
    J. E. Dennis Jr. and H. Wolkowicz,Sizing and least-change secant methods SIAM J. Numer. Anal., 30 (1993), pp. 1291–1314.Google Scholar
  16. 16.
    J. M. Donato and T. C. Chan,Fourier analysis of incomplete factorization preconditioners for three-dimensional anisotropic problems SIAM J. Sci. Stat. Comput., 13 (1992), pp. 319–338.Google Scholar
  17. 17.
    P. F. Dubois, A. Greenbaum, and G. H. Rodrigue,Approximating the inverse of a matrix for use in iterative algorithms on vector processors Computing, 22 (1979), pp. 257–268.Google Scholar
  18. 18.
    A. Greenbaum,Comparison of splittings used with the conjugate gradient algorithm Numer. Math., 33 (1979), pp. 181–194.Google Scholar
  19. 19.
    A. Greenbaum and G. H. Rodrigue,Optimal preconditioners of a given sparsity pattern BIT, 29 (1989), pp. 610–634.Google Scholar
  20. 20.
    I. Gustafsson,A class of first order factorization methods BIT, 18 (1978), pp. 142–156.Google Scholar
  21. 21.
    A. Jennings,Influence of the eigenvalue spectrum on the convergence rate of the conjugate gradient method J. Inst. Maths. Applics. 20 (1977), pp. 61–72.Google Scholar
  22. 22.
    O. G. Johnson, C. A. Micchelli, and G. Paul,Polynomial preconditioners for conjugate gradient calculations SIAM J. Numer. Anal. 20 (1983), pp. 362–376.Google Scholar
  23. 23.
    I. E. Kaporin,New convergence results and preconditioning strategies for the conjugate gradient method, Preprint, Dept. of Comp. Math. and Cyb., Moscow State University, 1992.Google Scholar
  24. 24.
    The Mathworks,Pro-Matlab User's Guide, The Mathworks, 1990.Google Scholar
  25. 25.
    J. A. Meijerink and H. A. van der Vorst,An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix Math. Comp., 31 (1977), pp. 148–162.Google Scholar
  26. 26.
    D. P. O'Leary,Yet another polynomial preconditioner for the conjugate gradient algorithm Linear Algebra Appl., 154/56 (1991), pp. 377–388.Google Scholar
  27. 27.
    G. Pini and G. Gambolati,Is a simple diagonal scaling the best preconditioner for conjugate gradients on supercomputers? Adv. Water Resources, 13 (1990), pp. 147–153.Google Scholar
  28. 28.
    W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling,Numerical Recipes in C. The Art of Scientific Computing, Cambridge University Press, 1988.Google Scholar
  29. 29.
    Z. Strakoš,On the real convergence rate of the conjugate gradient method Linear Algebra Appl., 154/56 (1991), pp. 535–549.Google Scholar
  30. 30.
    A. van der Sluis and H. A. van der Vorst,The rate of convergence of conjugate gradients Numer. Math., 48 (1986), pp. 543–560.Google Scholar
  31. 31.
    R. Winther,Some superlinear convergence results for the conjugate gradient method SIAM J. Numer. Anal., 17 (1980), pp. 14–17.Google Scholar

Copyright information

© the BIT Foundation 1994

Authors and Affiliations

  • A. M. Bruaset
    • 1
  • A. Tveito
    • 1
  1. 1.SINTEFOsloNorway

Personalised recommendations