Skip to main content

Advertisement

SpringerLink
Log in

Fast parallel solution of boundary integral equations and related problems

  • Regular article
  • Published:
Computing and Visualization in Science

Abstract

This article is concerned with the efficient numerical solution of Fredholm integral equations on a parallel computer with shared or distributed memory. Parallel algorithms for both, the approximation of the discrete operator by hierarchical matrices using adaptive cross approximation (ACA) and the parallel matrix-vector multiplication of such matrices by a vector, are presented. The first algorithm has a complexity of order p -1 N log2d-1 N, while the latter is of order p -1 N logd N, where N, d and p are the number of unknowns, the spatial dimension and the number of processors, respectively. The approximant needs Ω(p -1 N logd N) units of storage on each processor.

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. Bebendorf, M.: Approximation of boundary element matrices. Numer. Math. 86(4), 565–589 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bebendorf, M.: Effiziente numerische Lösung von Randintegralgleichungen unter Verwendung von Niedrigrang-Matrizen. PhD thesis, Universität Saarbrücken, 2000. dissertation.de, Verlag im Internet (2001) ISBN 3-89825-183--7.

  3. Bebendorf, M.: Efficient Galerkin BEM using ACA. Technical report, Max-Planck-Institute (2004)

  4. Bebendorf, M., Rjasanow, S.: Adaptive low-rank approximation of collocation matrices. Computing. 70(1), 1–24 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  5. Butenhof, D.R.: Programming with POSIX threads. Addison-Wesley (1997)

  6. Cheng, H., Greengard, L.F., Rokhlin, V.: A fast adaptive multipole algorithm in three dimensions. J. Comput. Phys. 155(2), 468–498 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  7. Dongarra, J., Demmel, J.: LAPACK: a portable high-performance numerical library for linear algebra. Supercomputer. 8, 33–38 (1991)

    Google Scholar 

  8. Goreinov, S.A., Tyrtyshnikov, E.E.: The maximal-volume concept in approximation by low-rank matrices. In Structured matrices in mathematics, computer science, and engineering, I (Boulder, CO, 1999) volume 280 of Contemp. Math. pp. 47–51. Amer. Math. Soc. Providence, RI (2001)

  9. Graham, R.L.: Bounds on multiprocessing timing anomalies. SIAM Journal of Applied Math. 17(2), 416–429 (1969)

    Article  MATH  Google Scholar 

  10. Greengard, L.F., Rokhlin, V.: A new version of the fast multipole method for the Laplace equation in three dimensions. In Acta numerica, 1997, vol. 6 of Acta Numer. pp. 229–269. Cambridge Univ. Press, Cambridge (1997)

  11. Hackbusch, W.: A sparse matrix arithmetic based on H-matrices. I. Introduction to H-matrices. Computing. 62(2), 89–108 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  12. Hackbusch, W., Khoromskij, B.N.: A sparse H-matrix arithmetic. II. Application to multi-dimensional problems. Computing. 64(1), 21–47 (2000)

    MATH  MathSciNet  Google Scholar 

  13. Hackbusch, W., Nowak, Z.P.: On the fast matrix multiplication in the boundary element method by panel clustering. Numer. Math. 54(4), 463–491 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  14. Kriemann, R.: Implementation and Usage of a Thread Pool based on POSIX Threads. MPI MIS Leipzig (2003) Report 2/2003.

  15. Langer, U., Pusch, D., Reitzinger, S.: Efficient preconditioners for boundary element matrices based on grey-box algebraic multigrid methods. Internat. J. Numer. Methods Engrg. 58(13), 1937–1953 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  16. McColl, W.F.: Scalable computing. In Computer Science Today: Recent Trends and Developments. vol. 1000 of LNCS, pp. 46–61. Springer-Verlag (1995)

  17. Olstad, B., Manne, F.: Efficient partitioning of sequences. IEEE Trans. Comput. 44 1322–1326 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  18. Rokhlin, V.: Rapid solution of integral equations of classical potential theory. J. Comput. Phys. 60(2), 187–207 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  19. Rokhlin, V.: Sparse diagonal forms for translation operators for the Helmholtz equation in two dimensions. Appl. Comput. Harmon. Anal. 5(1), 36–67 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  20. Rain, O., Kurz, S., Rjasanow, S.: The adaptive cross approximation technique for the 3d boundary element method. IEEE Transaction on Magnetics. 38(2), 421–424 (2002)

    Article  Google Scholar 

  21. Tarjan, R.E.: Depth-first search and linear graph algorithms. SIAM Journal on Computing. 1(2), 146–160 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  22. Tyrtyshnikov, E.E.: Mosaic-skeleton approximations. Calcolo. 33(1-2):47–57 (1998), 1996. Toeplitz matrices: structures, algorithms and applications Cortona (1996)

    Article  MathSciNet  Google Scholar 

  23. Tyrtyshnikov, E.E.: Incomplete cross approximation in the mosaic-skeleton method. Computing. 64(4), 367–380 (2000) International GAMM-Workshop on Multigrid Methods (Bonn, 1998)

    Article  MATH  MathSciNet  Google Scholar 

  24. Valiant, L.G.: A bridging model for parallel computation. Communications of the ACM. 33(8), 103–111 (1990)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fakultät für Mathematik und Informatik, University of Leipzig, Augustusplatz 10/11, 04109, Leipzig, Germany

    M. Bebendorf

  2. Max-Planck-Institute for Mathematics in the Sciences, Inselstr. 22--26, 04103, Leipzig, Germany

    R. Kriemann

Authors
  1. M. Bebendorf
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. Kriemann
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Bebendorf.

Additional information

Dedicated to George C. Hsiao on the occasion of his 70th birthday.

Mathematics Subject Classification (2000)65D05 65D15 65F05 65F30

Communicated by: U. Langer

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bebendorf, M., Kriemann, R. Fast parallel solution of boundary integral equations and related problems. Comput. Visual Sci. 8, 121–135 (2005). https://doi.org/10.1007/s00791-005-0001-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00791-005-0001-x

Key words

Search

Navigation