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Modified block-approximate factorization strategies

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Summary

Two variants of modified incomplete block-matrix factorization with additive correction are proposed for the iterative solution of large linear systems of equations. Both rigorous theoretical support and numerical evidence are given displaying their efficiency when applied to discrete second order partial differential equations (PDEs), even in the case of quasi-singular problems.

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References

  1. Axelsson, O. (1972): A generalized SSOR method. BIT13, 443–467

    Google Scholar 

  2. Axelsson, O. (1989): On the eigenvalue distribution of relaxed incomplete factorization methods and the rate of convergence of preconditioned conjugate gradient method. XV National summer school on Application of Mathematics in Engineering, August 23–31, 1989, Varna, Bulgaria

  3. Axelsson, O., Barker, V.A.: Finite Element Solutions of Boundary Value Problems: Theory and Computation. Academic Press, New York, 1984

    Google Scholar 

  4. Axelsson, O., Eijkhout, V. (1989): Vectorizable preconditioners for elliptic difference equations in three space dimensions. J. Comput. Appl. Math.27, 299–321

    Google Scholar 

  5. Axelsson, O., Lindskog, G. (1986): On the eigenvalue distribution of a class of preconditioning methods. Numer. Math.48, 479–498

    Google Scholar 

  6. Beauwens, R. (1987): Lower eigenvalue bounds for pencils of matrices. Lin Alg. Appl.85, 101–119

    Google Scholar 

  7. Beauwens, R. (1985): On Axelsson's perturbations. Lin. Alg. Appl.68, 221–242

    Google Scholar 

  8. Beauwens, R. (1990): Modified incomplete factorization strategies. In: Axelsson, O., Kolotilina, L. eds., Preconditioned Conjugate Gradient Methods. Lectures Notes in Mathematics No. 1457, Springer, Berlin Heidelberg New York, pp. 1–16

    Google Scholar 

  9. Beauwens, R., Ben Bouzid, M. (1987): On sparse block factorization iterative methods. SIAM J. Numer. Anal.24, 1066–1076

    Google Scholar 

  10. Beauwens, R., Ben Bouzid, M. (1988): Existence and conditioning properties of sparse approximate block factorizations. SIAM J. Numer. Anal.25, 941–956

    Google Scholar 

  11. Beauwens, R., Wilmet, R. (1989): Conditioning analysis of positive definite matrices by approximate factorizations. J. Comput. Appl. Math.26, 257–269

    Google Scholar 

  12. Berman, A., Plemmons, R.J.: Nonnegative Matrices in the Mathematical Sciences. Academic Press, New York, 1979

    Google Scholar 

  13. Concus, P., Golub, G.H., Meurant, G. (1985): Block preconditioning for the conjugate gradient method. SIAM J. Sci. Statist. Comput.6, 220–252

    Google Scholar 

  14. George, A., Liu, J.W. (1981): Computer Solution of Large Sparse Positive Definite Systems. Prentice-Hall, Englewood Cliffs, N.J.

    Google Scholar 

  15. Goovaerts, D. (1990): Domain Decomposition Methods for Elliptic Partial Differential Equations. PhD Thesis, Katholieke Universiteit Leuven, Department of Computer Science, Leuven, Belgium

    Google Scholar 

  16. Gustafsson, I. (1983): Modified incomplete Cholesky (MIC) Methods. In: Evans, D.J. ed., Preconditioning Methods, Theory and Applications Gordon and Breach, New York London Paris, pp. 265–293

    Google Scholar 

  17. Harary, F. (1969): Graph Theory. Addison-Wesley, Reading

    Google Scholar 

  18. Kershaw, D.S. (1978): The incomplete Choleski-conjugate gradient method for the iterative solution of systems of linear equations. J. Comput. Phys.26, 43–65

    Google Scholar 

  19. Kettler, R. (1987): Linear multigrid method for numerical reservoir simulation. Ph. D. Thesis, University of Technology, Dept. of Technical Math. and Informatics, Delft

    Google Scholar 

  20. Magolu, M.M. (1991): Conditioning analysis, of sparse block approximate factorizations. Appl. Numer. Math.8, 25–42

    Google Scholar 

  21. Magolu, M.M.: Analytical bounds for block approximate factorization methods. Lin. Alg. Appl. (submitted)

  22. Magolu, M.M., Notay, Y. (1991): On the conditioning analysis of block approximate factorization methods. Lin. Alg. Appl.154–156, 583–599

    Google Scholar 

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

    Google Scholar 

  24. Notay, Y. (1989): Incomplete factorizations of singular linear systems. BIT29, 682–702

    Google Scholar 

  25. Notay, Y. (1991): Conditioning analysis of modified block incomplete factorizations. Lin. Alg. Appl.154–156, 711–722

    Google Scholar 

  26. Sonneveld, P. (1989): A fast Lanczos type solver for nonsymmetric linear system. SIAM J. Sci. Statist. Comput.10, 36–52

    Google Scholar 

  27. Van der Sluis, A., Van der Vorst, H.A. (1986): The rate of convergence of conjugate gradients. Numer. Math.48, 543–560

    Google Scholar 

  28. Van der Vorst, H.A. (1990): The convergence behaviour of preconditioned CG and CG-S. In: Axelsson, O., Kolotilina, L. eds., Preconditioned Conjugate Gradient Methods. Lectures Notes in Mathematics, No. 1457. Springer, Berlin Heidelberg New York, pp. 126–136

    Google Scholar 

  29. Varga, R.S. (1962): Matrix Iterative Analysis. Prentice-Hall, Englewood Cliffs, N.J.

    Google Scholar 

  30. Wittum, G. (1989): On the robustness of ILU-smoothing. SIAM J. Sci. Stat. Comput.10, 699–717

    Google Scholar 

  31. Wittum, G. (1989): Linear, iterations as smoothers in Multigrid methods: Theory with applications to incomplete decompositions. Impact Comput. Sci. Engrg.1, 180–215

    Google Scholar 

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Research supported by the A.B.O.S. (A.G.C.D.) under project 11, within the co-operation between Belgium and Zaire

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Magolu, MM. Modified block-approximate factorization strategies. Numer. Math. 61, 91–110 (1992). https://doi.org/10.1007/BF01385499

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  • DOI: https://doi.org/10.1007/BF01385499

Mathematics Subject Classification (1991)

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