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A Class of Nested Iteration Schemes for Linear Systems with a Coefficient Matrix with a Dominant Positive Definite Symmetric Part

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Abstract

We present a class of nested iteration schemes for solving large sparse systems of linear equations with a coefficient matrix with a dominant symmetric positive definite part. These new schemes are actually inner/outer iterations, which employ the classical conjugate gradient method as inner iteration to approximate each outer iterate, while each outer iteration is induced by a convergent and symmetric positive definite splitting of the coefficient matrix. Convergence properties of the new schemes are studied in depth, possible choices of the inner iteration steps are discussed in detail, and numerical examples from the finite-difference discretization of a second-order partial differential equation are used to further examine the effectiveness and robustness of the new schemes over GMRES and its preconditioned variant. Also, we show that the new schemes are, at least, comparable to the variable-step generalized conjugate gradient method and its preconditioned variant.

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References

  1. O. Axelsson, A generalized SSOR method, BIT 12 (1972) 443–467.

    Google Scholar 

  2. O. Axelsson, A generalized conjugate gradient, least square method, Numer. Math. 51 (1987) 209–227.

    Google Scholar 

  3. O. Axelsson, A restarted version of a generalized preconditioned conjugate gradient method, Commun. Appl. Numer. Methods 4 (1988) 521–530.

    Google Scholar 

  4. O. Axelsson, Iterative Solution Methods (Cambridge Univ. Press, Cambridge, 1994).

    Google Scholar 

  5. O. Axelsson and V.A. Barker, Finite Element Solution of Boundary Value Problems: Theory and Computation (Academic Press, New York, 1984).

    Google Scholar 

  6. O. Axelsson and M. Nikolova, Avoiding slave points in adaptive refinement procedures for convection-diffusion problems in 2D, Computing 62 (1998) 331–357.

    Google Scholar 

  7. O. Axelsson and P.S. Vassilevski, A black box generalized conjugate gradient solver with inner iterations and variable-step preconditioning, SIAM J. Matrix Anal. Appl. 12 (1991) 625–644.

    Google Scholar 

  8. O. Axelsson and P.S. Vassilevski, Construction of variable-step preconditioners for inner-outer iteration methods, in: Iterative Methods in Linear Algebra, eds. R. Beauwens and P. de Groen (Elsevier Science, Amsterdam, 1992) pp. 1–14.

    Google Scholar 

  9. O. Axelsson and P.S. Vassilevski, Variable-step multilevel preconditioning methods, I: Self-adjoint and positive definite elliptic problems, Numer. Linear Algebra Appl. 1 (1994) 75–101.

    Google Scholar 

  10. Z.Z. Bai, A class of modified block SSOR preconditioners for symmetric positive definite systems of linear equations, Adv. Comput. Math. 10 (1999) 169–186.

    Google Scholar 

  11. Z.Z. Bai, Sharp error bounds of some Krylov subspace methods for non-Hermitian linear systems, Appl. Math. Comput. 109(2/3) (2000) 273–285.

    Google Scholar 

  12. Z.Z. Bai, Modified block SSOR preconditioners for symmetric positive definite linear systems, Ann. Oper. Res. 103 (2001) 263–282.

    Google Scholar 

  13. Z.Z. Bai, I.S. Duff and A.J. Wathen, A class of incomplete orthogonal factorization methods. I: Methods and theories, BIT 41 (2001) 53–70.

    Google Scholar 

  14. Z.Z. Bai and S.L. Zhang, A regularized conjugate gradient method for symmetric positive definite system of linear equations, J. Comput. Math. 24 (2002) 473–448.

    Google Scholar 

  15. A. Berman and R.J. Plemmons, Non-Negative Matrices in the Mathematical Sciences, 3rd ed. (Academic Press, New York, 1994).

    Google Scholar 

  16. J.W. Daniel, The conjugate gradient method for linear and nonlinear operator equations, SIAM J. Numer. Anal. 4 (1967) 10–26.

    Google Scholar 

  17. R.S. Dembo, S.C. Eisenstat and T. Steihaug, Inexact Newton methods, SIAM J. Numer. Anal. 19 (1982) 400–408.

    Google Scholar 

  18. I.S. Duff, A.M. Erisman and J.K. Reid, Direct Methods for Sparse Matrices (Oxford Univ. Press, London, 1986).

    Google Scholar 

  19. L.P. Franca, S.L. Frey and T.J.R. Hughes, Stabilised finite element methods. I: Application to the advective-diffusive model, Comput. Methods Appl. Mech. Engrg. 95 (1992) 253–276.

    Google Scholar 

  20. G.H. Golub and M.L. Overton, Convergence of a two-stage Richardson iterative procedure for solving systems of linear equations, in: Proc. of the 9th Biennial Conf. on Numerical Analysis, Dundee, Scotland, 1981, ed. G.A. Watson, Lecture Notes in Mathematics, Vol. 912 (Springer, New York) pp. 128–139.

    Google Scholar 

  21. G.H. Golub and M.L. Overton, The convergence of inexact Chebyshev and Richardson iterative methods for solving linear systems, Numer. Math. 53 (1988) 571–593.

    Google Scholar 

  22. G.H. Golub and C.F. Van Loan, Matrix Computations, 3rd ed. (Johns Hopkins Univ. Press, Baltimore/London, 1996).

    Google Scholar 

  23. G.H. Golub and Q. Ye, Inexact preconditioned conjugate gradient method with inner-outer iteration, SIAM J. Sci. Comput. 21 (2000) 1305–1320.

    Google Scholar 

  24. A. Greenbaum, Iterative Methods for Solving Linear Systems (SIAM, Philadelphia, PA, 1997).

    Google Scholar 

  25. M.R. Hestenes and E.L. Stiefel, Methods of conjugate gradients for solving linear systems, J. Res. National Bureau Standards B 49 (1952) 409–436.

    Google Scholar 

  26. T.A. Manteuffel, An incomplete factorization technique for positive definite linear systems, Math. Comp. 34 (1980) 473–497.

    Google Scholar 

  27. 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) 148–162.

    Google Scholar 

  28. K.W. Morton, Numerical Solution of Convection-Diffusion Problems, Applied Mathematics and Mathematical Computation, Vol. 12 (Chapman and Hall, London, 1996).

    Google Scholar 

  29. N.K. Nichols, On the convergence of two-stage iterative processes for solving linear equations, SIAM J. Numer. Anal. 10 (1973) 460–469.

    Google Scholar 

  30. N.K. Nichols, On the local convergence of certain two step iterative procedures, Numer. Math. 24 (1975) 95–101.

    Google Scholar 

  31. Y. Saad, A flexible inner-outer preconditioned GMRES algorithm, SIAM J. Sci. Statist. Comput. 14 (1993) 461–469.

    Google Scholar 

  32. Y. Saad, ILUT: A dual threshold incomplete ILU factorization, Numer. Linear Algebra Appl. 1 (1994) 387–402.

    Google Scholar 

  33. Y. Saad, Iterative Methods for Sparse Linear Systems (PWS Publishing, Boston, 1996).

    Google Scholar 

  34. Y. Saad and M.H. Schultz, GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Statist. Comput. 7 (1986) 856–869.

    Google Scholar 

  35. A. van der Sluis and H.A. van der Vorst, The rate of convergence of conjugate gradients, Numer. Math. 48 (1986) 543–560.

    Google Scholar 

  36. H.A. van der Vorst, Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of non-symmetric linear systems, SIAM J. Sci. Statist. Comput. 13 (1992) 631–644.

    Google Scholar 

  37. H.A. van der Vorst and C. Vuik, GMRESR: A family of nested GMRES methods, Numer. Linear Algebra Appl. 1 (1994) 369–386.

    Google Scholar 

  38. R.S. Varga, Matrix Iterative Analysis (Prentice-Hall, Englewood Cliffs, NJ, 1962).

    Google Scholar 

  39. X.G. Wang, K.A. Gallivan and R. Bramley, CIMGS: A incomplete orthogonalization preconditioner, SIAM J. Sci. Comput. 18 (1997) 516–536.

    Google Scholar 

  40. D.M. Young, Iterative methods for solving partial differential equations of elliptic type, Ph.D. thesis, Harvard University (1950).

  41. D.M. Young, Iterative Solution of Large Linear Systems (Academic Press, New York, 1971).

    Google Scholar 

  42. S.L. Zhang, Generalized product-type methods based on Bi-CG for solving nonsymmetric linear systems, SIAM J. Sci. Comput. 18 (1997) 537–551.

    Google Scholar 

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Authors and Affiliations

  1. Faculty of Mathematics and Informatics, Catholic University, NL-6525 ED, Nijmegen, The Netherlands

    Owe Axelsson

  2. State Key Laboratory of Scientific/Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing, 100080, P.R. China

    Zhong-Zhi Bai & Shou-Xia Qiu

Authors
  1. Owe Axelsson
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  2. Zhong-Zhi Bai
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  3. Shou-Xia Qiu
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Axelsson, O., Bai, ZZ. & Qiu, SX. A Class of Nested Iteration Schemes for Linear Systems with a Coefficient Matrix with a Dominant Positive Definite Symmetric Part. Numerical Algorithms 35, 351–372 (2004). https://doi.org/10.1023/B:NUMA.0000021766.70028.66

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