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Circulant block-factorization preconditioning of anisotropic elliptic problems

Zirkulante Präkonditionierung durch Block-Faktorisierung bei anisotropen elliptischen Problemen

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Abstract

The recently introduced circulant block-factorization preconditioners are studied in this paper. The general approach is first formulated for the case of block-tridiagonal sparse matrices. Then an estimate of the condition number of the preconditioned matrix for a model anisotropic Dirichlet boundary value problem is derived in the formκ<√2ε(n+1)+2, whereN=n 2 is the size of the discrete problem, andε stands for the ratio of the anisotropy. Various numerical tests demonstrating the behavior of the circulant block-factorization preconditioners for anisotropic problems are presented.

Zusammenfassung

In dieser Arbeit werden die kürzlich eingeführten zirkulanten Präkonditionierer untersucht, die aus Block-Faktorisierungen herrühren. Der allgemeine Zugang ist für schwach besetzte Block-Tridiagonalmatrizen formuliert. Es wird die Kondition der präkonditionierten Systemmatrix für ein Modellproblem, eine anisotrope Dirichlet-Randwertaufgabe, durch die Größeκ<√2ε(n+1)+2 abgeschätzt. Dabei stelltN=n 2 die Größe des diskretisierten Problems undε das Anisotropieverhältnis dar. Verschiedene numerische Experimente werden vorgestellt, die das Verhalten der Präkonditionierung beim Modellproblem aufzeigen.

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References

  1. Axelsson, O.: A survey of vectorizable preconditioning methods for large scale finite element matrices. In: Colloquium topics in applied numerical analysis (Verwer, J.G. ed.), pp. 21–47. Syllabus 4, Center of Mathematics and Informatics (CMI): Amsterdam 1983.

    Google Scholar 

  2. Axelsson, O., Barker, V. A.: Finite element solution of boundary value problems: Theory and Computations. Orlando: Academic Press 1983.

    Google Scholar 

  3. Axelsson, O., Brinkkemper, S. Il’in, V. P.: On some versions of incomplete block-matrix factorization methods. Lin. Alg. Appl.,38, 3–15 (1984).

    Article  Google Scholar 

  4. Axelsson, O., Polman, B.: On approximate factorization methods for block-matrices suitable for vector and parallel processors. Lin. Alg. Appl.77, 3–26 (1986).

    Article  MATH  Google Scholar 

  5. Axelsson, O., Eijkhout, V. L.: Robust vectorizable preconditioners for three-dimensional elliptic difference equations with anisotropy. In: Algorithms and applications on vector and parallel computers (te Riele, H.J.J., Dekker, Th.J. van der Vorst, H. eds.), pp. 279–306. Amsterdam: North-Holland 1987.

    Google Scholar 

  6. Chan, R. H., Chan, T. F.: Circulant preconditioners for elliptic problems. J. Num. Lin. Alg. Appl.1, 77–101 (1992).

    Google Scholar 

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

    Article  MATH  Google Scholar 

  8. Duff, I. S., Meurant, G. A.: The effect of ordering on preconditioned conjugate gradients. BIT29, 635–657 (1989).

    Article  MATH  Google Scholar 

  9. Golub, G. H., Van Loan, C. F.: Matrix computations, 2nd ed. Johns Hopkins Univ. Press: Baltimore 1989.

    MATH  Google Scholar 

  10. Gunn, J. E.: The solution of difference equations by semi-explicit iterative techniques. SIAM J. Num. Anal.2, 24–45 (1965).

    Article  Google Scholar 

  11. Gustafsson, I.: A class of first-order factorization methods. BIT18, 142–156 (1978).

    Article  MATH  Google Scholar 

  12. Hackbusch, W.: The frequency decomposition multi-grid method. 1. Application to anisotropic equations. Numer. Math.56, 219–245 (1989).

    Article  Google Scholar 

  13. Holmgren, S., Otto, K.: Iterative solution methods for block-tridiagonal systems of equations. SIAM J. Matr. Anal. Appl.13, 863–886 (1992).

    Article  MATH  Google Scholar 

  14. Huckle, T.: Some aspects of circulant preconditioners. SIAM J. Sci. Comput.14, 531–541 (1993).

    Article  MATH  Google Scholar 

  15. Van Loan, C.: Computational frameworks for the fast Fourier transform. Philadelphia: SIAM 1992.

    MATH  Google Scholar 

  16. Lirkov, I., Margenov, S.: On circulant preconditioning of elliptic problems in L-shaped domain. In: Advances in numerical methods and applications (Dimov, I. T., Sendov, Bl., Vassilevski, P. S., eds.), pp. 136–145. Singapore: World Scientific 1994.

    Google Scholar 

  17. Lirkov, I., Margenov, S., Vassilevski, P. S.: Circulant block-factorization preconditioners for elliptic problems. Computing53, 59–74 (1994).

    Article  MATH  Google Scholar 

  18. Margenov, S. D., Lirkov, I. T.: Preconditioned conjugate gradient iterative algorithms for transputer based systems. In: Parallel and distributed processing (Boyanov, K., ed.), pp. 406–415. Sofia: Bulgazian Academy of Sciences 1993.

    Google Scholar 

  19. Margenov, S. D., Vassilevski, P. S.: Algebraic multilevel preconditioning of anisotropic elliptic problems. SIAM J. Sci. Comp.15, 1026–1037 (1994).

    Article  MATH  Google Scholar 

  20. Meurant, G.: A review on the inverse of symmetric tridiagonal and block tridiagonal matrices. SIAM J. Matrix Anal. Appl.13, 707–728 (1992).

    Article  MATH  Google Scholar 

  21. Pester, M., Rjasanow, S.: A parallel version of the preconditioned conjugate gradient method for boundary element equations. Num. Lin. Alg. Appl.2, 1–16 (1995).

    Article  MATH  Google Scholar 

  22. Pester, M., Rjasanow, S.: A parallel preconditioned iterative realization of the panel method in 3D. Num. Lin. Alg. Appl.3, 65–80 (1996).

    Article  MATH  Google Scholar 

  23. Strang, G.: A proposal for Toeplitz matrix calculations. Stud. Appl. Math.74, 171–176 (1986).

    MATH  Google Scholar 

  24. Tyrtyshnikov, E.: Optimal and superoptimal circulant preconditioners. SIAM J. Matrix Anal.13, 459–473 (1992).

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Center for Informatics and Computer Technology, Acad. G. Bonchev str., Bl. 25-A, 1113, Sofia, Bulgaria

    I. D. Lirkov & S. D. Margenov

  2. Institute of Mathematics and Informatics, Acad. G. Bouchev str., Bl. 8, 1113, Sofia, Bulgaria

    L. T. Zikatanov

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  1. I. D. Lirkov
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  2. S. D. Margenov
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  3. L. T. Zikatanov
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Lirkov, I.D., Margenov, S.D. & Zikatanov, L.T. Circulant block-factorization preconditioning of anisotropic elliptic problems. Computing 58, 245–258 (1997). https://doi.org/10.1007/BF02684392

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  • DOI: https://doi.org/10.1007/BF02684392

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