Multigrid preconditioners for anisotropic space-fractional diffusion equations

Abstract

We focus on a two-dimensional time-space diffusion equation with fractional derivatives in space. The use of Crank-Nicolson in time and finite differences in space leads to dense Toeplitz-like linear systems. Multigrid strategies that exploit such structure are particularly effective when the fractional orders are both close to 2. We seek to investigate how structure-based multigrid approaches can be efficiently extended to the case where only one of the two fractional orders is close to 2, i.e., when the fractional equation shows an intrinsic anisotropy. Precisely, we design a multigrid (block-banded–banded-block) preconditioner whose grid transfer operator is obtained with a semi-coarsening technique and that has relaxed Jacobi as smoother. The Jacobi relaxation parameter is estimated by using an automatic symbol-based procedure. A further improvement in the robustness of the proposed multigrid method is attained using the V-cycle with semi-coarsening as smoother inside an outer full-coarsening. Several numerical results confirm that the resulting multigrid preconditioner is computationally effective and outperforms current state of the art techniques.

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

References

  1. 1.

    Aricò, A., Donatelli, M., Serra-capizzano, S.: V-cycle optimal convergence for certain (multilevel) structured linear systems. SIAM J. Matrix Anal. Appl. 26, 186–214 (2004)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Bai, J., Feng, X.: Fractional-order anisotropic diffusion for image denoising. IEEE Tran. on Image Process. 16, 2492–2502 (2007)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Bini, D., Capovani, M., Menchi, O.: Metodi Numerici per l’Algebra Lineare. Zanichelli, Bologna (1988)

    Google Scholar 

  4. 4.

    Defterli, O., D’Elia, M., Du, Q., Gunzburger, M., Lehoucq, R., Meerschaert, M.M.: Fractional diffusion on bounded domains,. Fract. Calc. Appl. Anal. 18(2), 342–360 (2015)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Donatelli, M.: An algebraic generalization of local Fourier analysis for grid transfer operators in multigrid based on Toeplitz matrices. Numer. Linear Algebra Appl. 17, 179–197 (2010)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Donatelli, M., Garoni, C., Manni, C., Serra-Capizzano, S., Speleers, H.: Two-grid optimality for Galerkin linear systems based on B-splines. Comput. Vis. Sci. 17, 119–133 (2015)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Donatelli, M., Mazza, M., Serra-Capizzano, S.: Spectral analysis and structure preserving preconditioners for fractional diffusion equations. J. Comput. Phys. 307, 262–279 (2016)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Donatelli, M., Mazza, M., Serra-Capizzano, S.: Spectral analysis and multigrid methods for finite volume approximations of space-fractional diffusion equations. SIAM J. Sci Comput. 40, A4007–A4039 (2018)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Fischer, R., Huckle, T.: Multigrid methods for anisotropic BTTB systems. Linear Algebra Appl. 417, 314–334 (2006)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Fischer, R., Huckle, T.: Multigrid solution techniques for anisotropic structured linear systems. Appl. Numer. Math. 58, 407–421 (2008)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Garoni, C., Serra-Capizzano, S.: Generalized Locally Toeplitz Sequences: Theory and Applications, Vol. I. Springer, Cham (2017)

    Google Scholar 

  12. 12.

    Hemker, P.W.: On the order of prolongations and restrictions in multigrid procedures. J. Comput. Appl. Math. 32(3), 423–429 (1990)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Jiang, Y., Xu, X.: Multigrid methods for space fractional partial differential equations. J. Comput. Phys. 302, 374–392 (2015)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Lei, S.L., Sun, H.W.: A circulant preconditioner for fractional diffusion equations. J. Comput. Phys. 242, 715–725 (2013)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Lin, X., Ng, M. K., Sun, H.: A splitting preconditioner for Toeplitz-Like linear systems arising from fractional diffusion equations. SIAM J. Matrix Anal. Appl. 38, 1580–1614 (2017)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Lin, X., Ng, M.K., Sun, H.: A multigrid method for linear systems arising from time-dependent two-dimensional space-fractional diffusion equations. J. Comput. Phys. 336, 69–86 (2017)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Liu, F., Zhuang, P., Turner, I., Anh, V., Burrage, K.: A semi-alternating direction method for a 2-D fractional FitzHugh–Nagumo monodomain model on an approximate irregular domain. J. Comput. Phys. 293, 252–263 (2015)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Moghaderi, H., Dehghan, M., Donatelli, M., Mazza, M.: Spectral analysis and multigrid preconditioners for two-dimensional space-fractional diffusion equations. J Comput. Phys. 350, 992–1011 (2017)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Oldham, K.B., Spanier, J.: Fractional Calculus: Theory and Applications, Differentiation and Integration to Arbitrary Order. Academic Press, New York (1974)

    Google Scholar 

  20. 20.

    Oosterlee, C.W.: The convergence of parallel multiblock multigrid methods. Appl. Numer. Math. 19, 115–128 (1995)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Pang, H., Sun, H.W.: Multigrid method for fractional diffusion equations. J. Comput. Phys. 231, 693–703 (2012)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Ruge, J W., Stüben, K.: Algebraic multigrid Mccormick, S.F. (ed.), Multigrid Methods, Front. Math. Appl. SIAM, 3, 73-130 (1987)

  23. 23.

    Tian, W., Zhou, H., Deng, W.: A class of second order difference approximations for solving space-fractional diffusion equations. Math. Comp. 84, 1703–1727 (2015)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Serra-Capizzano, S.: Convergence analysis of two-grid methods for elliptic Toeplitz and PDEs matrix-sequences. Numer. Math. 92, 433–465 (2002)

    MathSciNet  Article  Google Scholar 

  25. 25.

    van Lent, J., Vandewalle, S.: Multigrid waveform relaxation for anisotropic partial differential equations. Numer. Algor. 31, 361–380 (2002)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Washio, T., Oosterlee, C.W.: Flexible multiple semi-coarsening for three-dimensional singularly perturbed problems. SIAM J. Sci. Comput. 19, 1646–1666 (1998)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Wang, H., Wang, K., Sircar, T.: A direct \(\mathcal {O}(n\log ^{2} {N})\) finite difference method for fractional diffusion equations. J. Comput. Phys. 229, 8095–8104 (2010)

    MathSciNet  Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Mariarosa Mazza.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work is partly supported by GNCS-INDAM (Italy).

Communicated by: Jan Hesthaven

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Donatelli, M., Krause, R., Mazza, M. et al. Multigrid preconditioners for anisotropic space-fractional diffusion equations. Adv Comput Math 46, 49 (2020). https://doi.org/10.1007/s10444-020-09790-2

Download citation

Keywords

  • Fractional diffusion equations
  • Toeplitz-like matrices
  • Spectral distribution
  • Preconditioning
  • Anisotropic multigrid methods

Mathematics Subject Classification (2010)

  • 35R11
  • 15B05
  • 65F15
  • 65F08
  • 65N55