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Robust incomplete factorization for nonsymmetric matrices

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Abstract

In this paper, a new incomplete LU factorization preconditioner for nonsymmetric matrices is being considered which is also breakdown-free (no zero pivots occurs) for positive definite matrices. To construct this preconditioner, only the information of matrix A is used and just one of the factors of the AINV process is computed. The L factor is extracted as a by-product of the AINV process. The pivots of the AINV process are used as diagonal entries of U. The new preconditioner has left and right-looking versions. To improve the efficiency of the preconditioner, we have used the inverse-based dropping strategies for both L and U factors. Numerical experiments show that the left-looking version of the preconditioner is significantly faster than its right-looking version in terms of preconditioning time and both are equally effective to reduce the number of iterations. Comparisons of the new preconditioner with AINV and ILUT preconditioners are also presented.

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References

  1. Benzi M., Tůma M.: A sparse approximate inverse preconditioner for nonsymmetric linear systems. SIAM J. Sci. Comput. 19(3), 968–994 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  2. Benzi M., Tůma M.: A robust incomplete factorization preconditioner for positive definite matrices. Numer. Linear Algebra Appl. 10, 385–400 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  3. Benzi, M., Tůma, M.: SPARSLAB software package (2009). http://www2.cs.cas.cz/tuma/sparslab.html. Accessed 17 November 2009

  4. Benzi M., Cullum J.K., Tůma M.: Robust approximate inverse preconditioning for the conjugate gradient method. SIAM J. Sci. Comput. 22, 1318–1332 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bollhöfer M.: A robust ILU based on monitoring the growth of the inverse factors. Linear Algebra Appl. 338(1–3), 201–218 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bollhöfer M., Saad Y.: On the relations between ILUs and factored approximate inverses. SIAM J. Matrix Anal. Appl. 24(1), 219–237 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bollhöfer M.: A robust and efficient ILU that incorporates the growth of the inverse triangular factors. SIAM J. Sci. Comput. 25(1), 86–103 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bollhöfer, M.: ILUPACK software package (2009). http://www-public.tu-bs.de/bolle/ilupack

  9. Cline A., Moler C.B., Wilkinson J.: An estimate for the condition number of a matrix. SIAM J. Numer. Anal. 16, 368–375 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  10. Davis, T.: University of Florida Sparse Matrix Collection (2009). http://www.cise.ufl.edu/research/sparse/matrices/. Accessed 17 November 2009

  11. Golub G.H., Van Loan C.F.: Matrix Computations, 3rd edn. The John Hopkins University Press, Baltimore (1996)

    MATH  Google Scholar 

  12. Freund R.W.: A transpose-free quasi-minimal residual algorithm for non-hermitian linear systems. SIAM J. Sci. Comput. 14(2), 470–482 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  13. Karypis G., Kumar V.: A fast and high quality multilevel scheme for partitioning irregular graphs. SIAM J. Sci. Comput. 20(1), 359–392 (1998)

    Article  MathSciNet  Google Scholar 

  14. Karypis, G.: METIS-Family of multilevel partitioning Algorithms/Karypis Lab. http://glaros.dtc.umn.edu/gkhome/views/metis

  15. Kharchenko S.A., Kolotilina L.Yu., Nikishin A.A., Yeremin A.Yu.: A robust AINV-type method for constructing sparse approximate inverse preconditioners in factored form. Numer. Linear Algebra Appl. 8, 165–179 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  16. Li N., Saad Y., Chow E.: Crout version of ILU for general sparse matrices. SIAM J. Sci. Comput. 25(2), 716–728 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  17. Matlab version 7.5.0.338 (R2007b), The Math works Inc

  18. Rafiei A., Toutounian F.: New breakdown-free variant of AINV method for nonsymmetric positive definite matrices. J. Comput. Appl. Math. 219(1), 72–80 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Rafiei A., Toutounian F.: Breakdown-free version of ILU factorization for nonsymmetric positive definite matrices. J. Comput. Appl. Math. 230, 699–705 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Rafiei, A., Bollhöfer, M.: Extension of inverse-based dropping techniques for ILU preconditioners. http://www.digibib.tu-bs.de/?docid=00035624

  21. Saad Y.: Iterative Methods for Sparse Linear Systems. PWS publishing, Boston (1996)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  23. Saad, Y.: SPARSKIT and sparse examples. NA Digest (1994). http://www-users.cs.umn.edu/saad/software. Accessed 17 November 2009

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

    Article  MathSciNet  MATH  Google Scholar 

  25. van der Vorst H.A.: Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of non-symmetric linear systems. SIAM J. Sci. Stat. Comput. 12, 631–644 (1992)

    Google Scholar 

  26. Zlatev Z.: Computational Methods for General Sparse Matrices. Kluwer, Dordrecht (1991)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Applied Mathematics, Sabzevar Tarbiat Moallem University, Sabzevar, Iran

    A. Rafiei

  2. Institute computational Mathematics, Technische Universität Braunschweig, 38106, Braunschweig, Germany

    M. Bollhöfer

Authors
  1. A. Rafiei
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  2. M. Bollhöfer
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Correspondence to A. Rafiei.

Additional information

Most part of the work was done when A. Rafiei was on his sabbatical at Technische Universität Braunschweig.

M. Bollhöfer’s work was supported by the grant of the DFG under the contract number BO 1680/2-1.

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Rafiei, A., Bollhöfer, M. Robust incomplete factorization for nonsymmetric matrices. Numer. Math. 118, 247–269 (2011). https://doi.org/10.1007/s00211-010-0336-1

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  • DOI: https://doi.org/10.1007/s00211-010-0336-1

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