Function-Based Algebraic Multigrid Method for the 3D Poisson Problem on Structured Meshes

  • Ali DorostkarEmail author
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 126)


Multilevel methods, such as Geometric and Algebraic Multigrid, Algebraic Multilevel Iteration, Domain Decomposition-type methods have been shown to be the methods of choice for solving linear systems of equations, arising in many areas of Scientific Computing. The methods, in particular the multigrid methods, have been efficiently implemented in serial and parallel and are available via many scientific libraries.

The multigrid methods are primarily used as preconditioners for various Krylov subspace iteration methods. They exhibit convergence that is independent or nearly independent on the number of degrees of freedom and can be tuned to be also robust with respect to other problem parameters. Since these methods utilize hierarchical structures, their parallel implementation might exhibit lesser scalability.

In this work we utilize a different framework to construct multigrid methods, based on an analytical function representation of the matrix, that keeps the amount of computation high and local, and reduces the memory requirements. This approach is particularly suitable for modern computer architectures. An implementation of the latter for the three-dimensional discrete Laplace operator is derived and implemented. The same function representation technology is used to construct smoothers of sparse approximate inverse type.


  1. 1.
    A. Aricó, M. Donatelli, S. Serra-Capizzano, V-cycle optimal convergence for certain (multilevel) structured linear systems. SIAM J. Matrix Anal. Appl. 26(1), 186–214 (2004)MathSciNetCrossRefGoogle Scholar
  2. 2.
    O. Axelsson, Iterative Solution Methods (Cambridge University Press, Cambridge, 1996)zbMATHGoogle Scholar
  3. 3.
    M. Donatelli, S. Serra-Capizzano, Multigrid methods for (multilevel) structured matrices associated to a symbol and related applications. Bollettino dell Unione Matematica Italiana 9(6), 319–347 (2013)MathSciNetzbMATHGoogle Scholar
  4. 4.
    M. Donatelli, S. Serra-Capizzano, D. Sesana, Multigrid methods for Toeplitz linear systems with different size reduction. BIT Numer. Math. 52(2), 305–327 (2012)MathSciNetCrossRefGoogle Scholar
  5. 5.
    M. Donatelli, M. Molteni, V. Pennati, S. Serra-Capizzano. Multigrid methods for cubic spline solution of two point (and 2D) boundary value problems. Appl. Numer. Math. 104(Supplement C), 15–29 (2016). Sixth International Conference on Numerical Analysis – Recent Approaches to Numerical Analysis: Theory, Methods and Applications (NumAn 2014)Google Scholar
  6. 6.
    M. Donatelli, A. Dorostkar, M. Mazza, M. Neytcheva, S. Serra-Capizzano, Function-based block multigrid strategy for a two-dimensional linear elasticity-type problem. Comput. Math. Appl. 74(5), 1015–1028 (2017). SI: SDS2016 – Methods for PDEsGoogle Scholar
  7. 7.
    M. Donatelli, C. Garoni, C. Manni, S. Serra-Capizzano, H. Speleers, Symbol-based multigrid methods for galerkin b-spline isogeometric analysis. SIAM J. Numer. Anal. 55(1), 31–62 (2017)MathSciNetCrossRefGoogle Scholar
  8. 8.
    G. Fiorentino, S. Serra-Capizzano, Multigrid methods for Toeplitz matrices. Calcolo 28(3–4), 283–305 (1991)MathSciNetCrossRefGoogle Scholar
  9. 9.
    G. Fiorentino, S. Serra-Capizzano, Multigrid methods for symmetric positive definite block Toeplitz matrices with nonnegative generating functions. SIAM J. Sci. Comput. 17(5), 1068–1081 (1996)MathSciNetCrossRefGoogle Scholar
  10. 10.
    C. Garoni, S. Serra-Capizzano, Generalized Locally Toeplitz Sequences: Theory and Applications, vol. I (Springer, 2017)Google Scholar
  11. 11.
    T. Huckle, A. Kallischko, Frobenius norm minimization and probing for preconditioning. Int. J. Comput. Math. 84(8), 1225–1248 (2007)MathSciNetCrossRefGoogle Scholar
  12. 12.
    S. Kharchenko, L. Y. Kolotilina, A. Nikishin, A.Y. Yeremin, A robust ainv-type method for constructing sparse approximate inverse preconditioners in factored form. Numer. Linear Algebra Appl. 8(3), 165–179 (2001)MathSciNetCrossRefGoogle Scholar
  13. 13.
    L.Y. Kolotilina, A.Y. Yeremin, Factorized sparse approximate inverse preconditionings I. theory. SIAM J. Matrix Anal. Appl. 14(1), 45–58 (1993)Google Scholar
  14. 14.
    A. Napov, Y. Notay, An algebraic multigrid method with guaranteed convergence rate. SIAM J. Sci. Comput. 34(2), A1079–A1109 (2012)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Y. Saad, Iterative Methods for Sparse Linear Systems (SIAM, Philadelphia, 2003)CrossRefGoogle Scholar
  16. 16.
    E. Turan, P. Arbenz, Large scale micro finite element analysis of 3d bone poroelasticity. Parallel Comput. 40(7), 239–250 (2014). 7th Workshop on Parallel Matrix Algorithms and ApplicationsGoogle Scholar
  17. 17.
    E.E. Tyrtyshnikov, A unifying approach to some old and new theorems on distribution and clustering. Linear Algebra Appl. 232, 1–43 (1996)MathSciNetCrossRefGoogle Scholar
  18. 18.
    J. van Lent, S. Vandewalle, Multigrid waveform relaxation for anisotropic partial differential equations. Numer. Algorithms 31(1), 361–380 (2002)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Information TechnologyUppsala UniversityUppsalaSweden

Personalised recommendations