BIT Numerical Mathematics

, Volume 45, Issue 4, pp 819–850 | Cite as

Applications of the Generalized Fourier Transform in Numerical Linear Algebra

  • Krister ÅhlanderEmail author
  • Hans Munthe-Kaas


Equivariant matrices, commuting with a group of permutation matrices, are considered. Such matrices typically arise from PDEs and other computational problems where the computational domain exhibits discrete geometrical symmetries. In these cases, group representation theory provides a powerful tool for block diagonalizing the matrix via the Generalized Fourier Transform (GFT). This technique yields substantial computational savings in problems such as solving linear systems, computing eigenvalues and computing analytic matrix functions such as the matrix exponential.

The paper is presenting a comprehensive self contained introduction to this field. Building upon the familiar special (finite commutative) case of circulant matrices being diagonalized with the Discrete Fourier Transform, we generalize the classical convolution theorem and diagonalization results to the noncommutative case of block diagonalizing equivariant matrices.

Applications of the GFT in problems with domain symmetries have been developed by several authors in a series of papers. In this paper we elaborate upon the results in these papers by emphasizing the connection between equivariant matrices, block group algebras and noncommutative convolutions. Furthermore, we describe the algebraic structure of projections related to non-free group actions. This approach highlights the role of the underlying mathematical structures, and provides insight useful both for software construction and numerical analysis. The theory is illustrated with a selection of numerical examples.

Key words

non commutative Fourier analysis equivariant operators block diagonalization 


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  1. 1.
    K. Åhlander and H. Munthe-Kaas, Eigenvalues for equivariant matrices, J. Comput. Appl. Math. (2005). Available online via ScienceDirect.Google Scholar
  2. 2.
    E. L. Allgower, K. Böhmer, K. Georg and R. Miranda, Exploiting symmetry in boundary element methods, SIAM J. Numer. Anal., 29 (1992), pp. 534–552.Google Scholar
  3. 3.
    E. L. Allgower and A. F. Fässler, Blockstructure and equivalence of matrices, in Aspects of Complex Analysis, Differential Geometry, Mathematical Physics and Applications, pp. 19–34, World Scientific, 1999.Google Scholar
  4. 4.
    E. L. Allgower and K. Georg, Exploiting symmetry in numerical solving, in Proceedings of the Seventh Workshop on Differential Equations and its Applications, C.-S. Chien, ed., Taichung, Taiwan, 1999.Google Scholar
  5. 5.
    E. L. Allgower, K. Georg and R. Miranda, Exploiting permutation symmetry with fixed points in linear equations, in Lectures in Applied Mathematics, E. L. Allgower, K. Georg and R. Miranda, eds., vol. 29, American Mathematical Society, pp. 23–36, Providence, RI, 1993.Google Scholar
  6. 6.
    E. L. Allgower, K. Georg, R. Miranda and J. Tausch, Numerical exploitation of equivariance, Z. Angew. Math. Mech., 78 (1998), pp. 185–201.Google Scholar
  7. 7.
    T. Beth, Generalized Fourier Transforms, in Trends in Computer Algebra, Lect. Notes Comput. Sci., vol. 296, pp. 92–118, Springer, Berlin, 1988.Google Scholar
  8. 8.
    M. Bonnet, Exploiting partial or complete geometrical symmetry in 3D symmetric Galerkin indirect BEM formulations, Int. J. Numer. Methods Eng., 57 (2003), pp. 1053–1083.Google Scholar
  9. 9.
    A. Bossavit, Symmetry, groups, and boundary value poblems. a progressive introduction to noncommutative harmonic analysis of partial differential equations in domains with geometrical symmetry, Comput. Methods Appl. Mech. Eng., 56 (1986), pp. 167–215.Google Scholar
  10. 10.
    A. Bossavit, Boundary value problems with symmetry and their approximation by finite elements, SIAM J. Appl. Math., 53 (1993), pp. 1352–1380.Google Scholar
  11. 11.
    A. Clausen, Fast Fourier transforms for metabelian groups, SIAM J. Comput., 18 (1989), pp. 55–63.Google Scholar
  12. 12.
    A. Clausen and U. Baum, Fast Fourier Transforms, Wissenschaftsverlag, Mannheim, 1993.Google Scholar
  13. 13.
    A. Clausen and M. Müller, Generating fast Fourier transforms of solvable groups, J. Symb. Comput., 37 (2004), pp. 137–156.Google Scholar
  14. 14.
    P. Diaconis, Group representations in probability and statistics, in Lecture Notes – Monograph Series, Institute of Mathematical Statistics, Hayward, CA, 1988.Google Scholar
  15. 15.
    P. Diaconis and D. N. Rockmore, Efficient computation of the Fourier transform on finite groups, J. Am. Math. Soc., 3 (1990), pp. 297–332.Google Scholar
  16. 16.
    C. C. Douglas and J. Mandel, Abstract theory for the domain reduction method, Computing, 48 (1992), pp. 73–96.Google Scholar
  17. 17.
    S. Egner and M. Puschel, Symmetry-based matrix factorization, J. Symb. Comput., 37 (2004), pp. 157–186.Google Scholar
  18. 18.
    A. F. Fässler and E. Stiefel, Group Theoretical Methods and Their Applications, Birkhäuser, Boston, 1992.Google Scholar
  19. 19.
    K. Georg and R. Miranda, Exploiting symmetry in solving linear equations, in Bifurcation and Symmetry, E. L. Allgower, K. Böhmer and M. Golubisky, eds., ISNM, vol. 104, pp. 157–168, Birkhäuser, Basel, 1992.Google Scholar
  20. 20.
    K. Georg and R. Miranda, Symmetry aspects in numerical linear algebra with applications to boundary element methods, in Lectures in Applied Mathematics, E. L. Allgower, K. Georg and R. Miranda, eds., vol. 29, pp. 213–228, American Mathematical Society, Providence, RI, 1993.Google Scholar
  21. 21.
    K. Georg and J. Tausch, A generalized Fourier transform for boundary element methods with symmetries, Technical report, Colorado State University, Ft. Collins, Colorado, 1994.Google Scholar
  22. 22.
    G. James and M. Liebeck, Representations and Characters of Groups, Cambridge University Press, 2nd edn., 2001. ISBN 052100392X.Google Scholar
  23. 23.
    M. Karpovsky, Fast Fourier transforms on finite non-abelian groups, IEEE Transact. Comput., 10 (1977), pp. 1028–1030.Google Scholar
  24. 24.
    M. Ljungberg and K. Åhlander, Generic Programming Aspects of symmetry exploiting numerical software, in Proceedings of European Congress on Computational Methods in Applied Sciences and Engineering, ECCOMAS 2004, Jyväskylä, 24–28 July 2004, P. Neittaanmäki et al., eds., 2004. Also available as Technical Report 2004-020 from the Department of Information Technology, Uppsala University.Google Scholar
  25. 25.
    J. S. Lomont, Applications of Finite Groups, Academic Press, New York, 1959.Google Scholar
  26. 26.
    D. K. Maslen, The efficient computation of Fourier transforms on the symmetric group, Math. Comput., 67 (1998), pp. 1121–1147.Google Scholar
  27. 27.
    D. K. Maslen and D. N. Rockmore, Generalized FFTs – a survey of some recent results, in Proceedings of the 1995 DIMACS Workshop on Groups and Computation, L. Finkelstein and W. Kantor, eds., June 1997, pp. 183–237.Google Scholar
  28. 28.
    D. K. Maslen and D. N. Rockmore, Separation of variables and the computation of Fourier transforms on finite groups, J. Am. Math. Soc., 10 (1997), pp. 169–214.Google Scholar
  29. 29.
    H. Munthe-Kaas, Symmetric FFTs; a general approach, in Topics in linear algebra for vector- and parallel computers, PhD thesis, NTNU, Trondheim, Norway, 1989. Available at Scholar
  30. 30.
    D. N. Rockmore, Fast Fourier analysis for abelian group extensions, Adv. Appl. Math., 11 (1990), pp. 164–204.Google Scholar
  31. 31.
    D. N. Rockmore, Fast Fourier-transforms for wreath–products, Appl. Comput. Harmonic Anal., 2 (1995), pp. 279–292.Google Scholar
  32. 32.
    D. N. Rockmore, Some applications of generalized FFTs, in Proceedings of the 1995 DIMACS Workshop on Groups and Computation, L. Finkelstein and W. Kantor, eds., June 1997, pp. 329–369.Google Scholar
  33. 33.
    J. P. Serre, Linear Representations of Finite Groups, Springer, 1977. ISBN 0387901906.Google Scholar
  34. 34.
    L. Stiller, Exploiting symmetry on parallel architectures, PhD thesis, Johns Hopkins University, 1995.Google Scholar
  35. 35.
    J. Tausch, Equivariant preconditioners for boundary element methods, SIAM Sci. Comput., 17 (1996), pp. 90–99.Google Scholar

Copyright information

© Springer-Verlag 2005

Authors and Affiliations

  1. 1.Department of Information TechnologyUppsala UniversityUppsalaSweden
  2. 2.Department of MathematicsUniversity of BergenBergenNorway

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