BIT Numerical Mathematics

, Volume 47, Issue 1, pp 213–237 | Cite as

Sparse generalized Fourier transforms



Block-diagonalization of sparse equivariant discretization matrices is studied. Such matrices typically arise when partial differential equations that evolve in symmetric geometries are discretized via the finite element method or via finite differences.

By considering sparse equivariant matrices as equivariant graphs, we identify a condition for when block-diagonalization via a sparse variant of a generalized Fourier transform (GFT) becomes particularly simple and fast.

Characterizations for finite element triangulations of a symmetric domain are given, and formulas for assembling the block-diagonalized matrix directly are presented. It is emphasized that the GFT preserves symmetric (Hermitian) properties of an equivariant matrix.

By simulating the heat equation at the surface of a sphere discretized by an icosahedral grid, it is demonstrated that the block-diagonalization is beneficial. The gain is significant for a direct method, and modest for an iterative method.

A comparison with a block-diagonalization approach based upon the continuous formulation is made. It is found that the sparse GFT method is an appropriate way to discretize the resulting continuous subsystems, since the spectrum and the symmetry are preserved.

Key words

non commutative Fourier analysis block-diagonalization equivariant operators sparse matrices finite differences finite elements 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    K. Åhlander, Supporting tensor symmetries in EinSum, Comput. Math. Appl., 45 (2003), pp. 789–803. Also available as Technical Report 212 from Dept. of Informatics, University of Bergen.Google Scholar
  2. 2.
    K. Åhlander and H. Munthe-Kaas, Applications of the generalized Fourier transform in numerical linear algebra, BIT Numerical Mathematics, (2005), pp. 819–850. Also available as Technical Report 2004-029, Dept. of Information Technology, Uppsala University.Google Scholar
  3. 3.
    K. Åhlander and H. Munthe-Kaas, Eigenvalues for equivariant matrices, J. Comput. Appl. Math., 192 (2006), pp. 89–99.CrossRefMathSciNetGoogle Scholar
  4. 4.
    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.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    E. L. Allgower and K. Georg, Numerical exploitation of symmetric structures in BEM, in Boundary Integral Methods – Numerical and Mathematical Aspects, M. Golberg, ed., pp. 289–306, Computational Mechanics Publications, Southampton, 1998.Google Scholar
  6. 6.
    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, pp. 23–36, American Mathematical Society, Providence, RI, 1993.Google Scholar
  7. 7.
    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.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    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.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    A. Bossavit, Boundary value problems with symmetry and their approximation by finite elements, SIAM J. Appl. Math., 53 (1993), pp. 1352–1380.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, vol. 15 of Texts in Applied Mathematics, 2nd ed., Springer, New York, 2002.Google Scholar
  11. 11.
    S. Egner and M. Puschel, Symmetry-based matrix factorization, J. Symb. Comput., 37 (2004), pp. 157–186.MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    F. Giraldo and T. Warburton, A nodal triangle-based spectral element method for the shallow water equations on the sphere, J. Comput. Phys., 207 (2005), pp. 129–150.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    G. Golub and C. van Loan, Matrix Computations, The John Hopkins University Press, Baltimore, MD, 1984.Google Scholar
  14. 14.
    D. Henriksson, Exploiting Symmetries when Solving the Heat Equation on a Spherical Icosahedral Grid, Master’s thesis, Uppsala University, Uppsala, Sweden, 2006.Google Scholar
  15. 15.
    C. Johnsson, Numerical solution of partial differential equations by the finite element method, Studentlitteratur, Lund, 1987.Google Scholar
  16. 16.
    E. Larsson, K. Åhlander, and A. Hall, Multi-dimensional option pricing using radial basis functions and the generalized Fourier transform, Techn. Rep. 2006-037, Department of Information Technology, Uppsala University, 2006.Google Scholar
  17. 17.
    J. S. Lomont, Applications of Finite Groups, Academic Press, New York, 1959.MATHGoogle Scholar
  18. 18.
    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, American Mathematical Society, Providence, RIGoogle Scholar
  19. 19.
    T. Minkwitz, Algorithms explained by symmetry, in Lect. Notes Comput. Sci., vol. 900, pp. 157–167, Springer, Berlin, Heidelberg, 1995.Google Scholar
  20. 20.
    H. Munthe-Kaas, On group Fourier analysis and symmetry preserving discretizations of PDEs, J. Phys. A, 39(19) (2006), pp. 5563–5584.MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    C. C. Paige and M. A. Saunders, Minres: Sparse symmetric equations, Scholar
  22. 22.
    J. P. Serre, Linear Representations of Finite Groups, Springer, New York, 1977. ISBN 0387901906.MATHGoogle Scholar
  23. 23.
    J. Tausch, Equivariant preconditioners for boundary element methods, SIAM Sci. Comput., 17 (1996), pp. 90–99.MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    A. Trønnes, Symmetries and Generalized Fourier Transforms Applied to Computing the Matrix Exponential, Master’s thesis, University of Bergen, Bergen, Norway, 2005.Google Scholar
  25. 25.
    J. Turski, Geometric Fourier analysis of the conformal camera for active vision, SIAM Review, 46 (2004), pp. 230–255.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science + Business Media B.V. 2007

Authors and Affiliations

  1. 1.Department of Information TechnologyUppsala UniversityUppsalaSweden

Personalised recommendations