Skip to main content
SpringerLink
Log in

On the Solution of Boundary Value Problems by Using Fast Generalized Approximate Inverse Banded Matrix Techniques

  • Published:
The Journal of Supercomputing Aims and scope Submit manuscript

Abstract

A class of finite difference schemes in conjunction with approximate inverse banded matrix techniques based on the concept of LU-type factorization procedures is introduced for computing fast explicit approximate inverses. Explicit preconditioned iterative schemes in conjunction with approximate inverse matrix techniques are presented for the efficient solution of banded linear systems. A theorem on the rate of convergence and estimates of the computational complexity required to reduce the L∞-norm of the error is presented. Applications of the method on linear and non-linear systems are discussed and numerical results are given.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. O. Axelsson. Iterative solution methods, Cambridge University Press, 1994.

  2. D. J. Evans. Preconditioning Methods: Theory and Applications, Gordon and Breach Science Publishers, 1983.

  3. G. A. Gravvanis. Explicit preconditioned generalized domain decomposition methods. I. J. Applied Mathematics, 4(1):57-71, 2000.

    Google Scholar 

  4. G. A. Gravvanis. Fast explicit approximate inverses for solving linear and non-linear finite difference equations. I. J. Differential Equations and Applics, 1(4):451-473, 2000.

    Google Scholar 

  5. G. A. Gravvanis. Explicit isomorphic iterative methods for solving arrow-type linear systems. I. J. Computer Math., 74(2):195-206, 2000.

    Google Scholar 

  6. G. A. Gravvanis. Generalized approximate inverse preconditioning for solving non-linear elliptic boundary-value problems. I. J. Applied Mathematics, 2(11):1363-1378, 2000.

    Google Scholar 

  7. G. A. Gravvanis. Domain decomposition approximate inverse preconditioning for solving fourth order equations. In H. Arabnia, ed., Proc. of Inter. Conference on Parallel and Distributed Processing Techniques and Applications, V. I, pp. 1-7. CSREA, 2000.

  8. G. A. Gravvanis. Explicit preconditioning conjugate gradient schemes for solving biharmonic equations. Engineering Computations, 17(2):154-165, 2000.

    Google Scholar 

  9. G. A. Gravvanis. Explicit approximate inverse finite element preconditioning for solving biharmonic equations. In M. Ingber, H. Power, and C.A. Brebbia, eds., Applications of High Performance Computing in Engineering VI, pp. 411-420. WIT Press, 2000.

  10. G. A. Gravvanis. Banded approximate inverse matrix techniques. Engineering Computations, 16(3):337-346, 1999.

    Google Scholar 

  11. G. A. Gravvanis. Parallel matrix techniques. In K. Papailiou, D. Tsahalis, J. Periaux, C. Hirsch, and M. Pandolfi, eds., Computational Fluid Dynamics 98, Vol. 1, pp. 472-477. Wiley, 1998.

  12. G. A. Gravvanis. An approximate inverse matrix technique for arrowhead matrices, I. J. Computer Math., 70:35-45, 1998.

    Google Scholar 

  13. G. A. Gravvanis. The rate of convergence of explicit approximate inverse preconditioning. I. J. Computer Math., 60:77-89, 1996.

    Google Scholar 

  14. G. A. Gravvanis. Explicit preconditioned methods for solving 3D BV problems by approximate inverse finite element matrix techniques. I. J. Computer Math., 56:77-93, 1995.

    Google Scholar 

  15. G. A. Gravvanis and E. A. Lipitakis. A three dimensional explicit preconditioned solver. Computer Mathematics with Applications, 32:111-131, 1996.

    Google Scholar 

  16. M. J. Grote and T. Huckle. Parallel preconditioning with sparse approximate inverses. SIAM J. Sci. Comput., 18:838-853, 1997.

    Google Scholar 

  17. I. Kaporin. New convergence results and preconditioning strategies for the conjugate gradient method. Numerical Linear Algebra Appl., 1:179-210, 1994.

    Google Scholar 

  18. E. A. Lipitakis and D. J. Evans. Numerical solution of non-linear elliptic boundary-value problems by isomorphic iterative methods. I. J. Computer Math., 20:261-282, 1986.

    Google Scholar 

  19. E. A. Lipitakis and G. A. Gravvanis. A class of explicit preconditioned conjugate gradient methods for solving large finite element systems. I. J. Computer Math., 44:189-206, 1992.

    Google Scholar 

  20. Y. Notay. On the convergence rate of the conjugate gradients in presence of rounding errors. Numer. Math., 65:301-317, 1993.

    Google Scholar 

  21. J. M. Ortega. Introduction to parallel and vector solution of linear systems, Plenum, 1989.

  22. Y. Saad. Iterative methods for sparse linear systems, PWS, 1996.

  23. A. van der Sluis and H. A. van der Vorst. The rate of convergence of conjugate gradients. Numer. Math., 48:543-560, 1986.

    Google Scholar 

  24. H. A. van der Vorst and C. Vuik. GMRESR: a family of nested GMRES methods. Numerical Linear Algebra Appl., 1(4):369-386, 1994.

    Google Scholar 

  25. C. Vuik and H. A. van der Vorst. A comparison of some GMRES-like methods. Linear Algebra Appl., 160:131-162, 1992.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Information and Communication Systems Engineering, University of the Aegean, GR 83200, Karlovasi, Samos, Greece

    George A. Gravvanis

Authors
  1. George A. Gravvanis
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gravvanis, G.A. On the Solution of Boundary Value Problems by Using Fast Generalized Approximate Inverse Banded Matrix Techniques. The Journal of Supercomputing 25, 119–129 (2003). https://doi.org/10.1023/A:1023936410006

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1023936410006

Search

Navigation