Numerical Algorithms

, Volume 74, Issue 4, pp 1061–1082 | Cite as

Solving mixed classical and fractional partial differential equations using short–memory principle and approximate inverses

  • Daniele BertacciniEmail author
  • Fabio Durastante
Original Paper


The efficient numerical solution of the large linear systems of fractional differential equations is considered here. The key tool used is the short–memory principle. The latter ensures the decay of the entries of the inverse of the discretized operator, whose inverses are approximated here by a sequence of sparse matrices. On this ground, we propose to solve the underlying linear systems by these approximations or by iterative solvers using sequence of preconditioners based on the above mentioned inverses.


Preconditioners Fractional calculus Iterative methods 

Mathematics Subject Classification (2010)

65F10 65F08 26A33 65Y10 


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  1. 1.
    Bellavia, S., Bertaccini, D., Morini, B.: Nonsymmetric preconditioner updates in Newton-Krylov methods for nonlinear systems. SIAM J. Sci. Comput. 33 (5), 2595–2619 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Benzi, M.: Preconditioning techniques for large linear systems: a survey. J. Comp. Phys. 182(2), 418–477 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Benzi, M., Bertaccini, D.: Approximate inverse preconditioning for shifted linear systems. BIT 43(2), 231–244 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bertaccini, D.: Efficient preconditioning for sequences of parametric complex symmetric linear systems. ETNA 18, 49–64 (2004)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bertaccini, D., Durastante, F.: Interpolating preconditioners for the solution of sequence of linear systems. Comp. Math. Appl. 72, 1118–1130 (2016)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bertaccini, D., Filippone, S.: Sparse approximate inverse preconditioners on high performance gpu platforms. Comp. Math. Appl. 71(3), 693–711 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bridson, R., Tang, W.-P.: Refining an approximate inverse. J. Comput. Appl. Math. 123(1), 293–306 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bridson, R., Tang, W.-P.: Multiresolution approximate inverse preconditioners. SIAM J. Sci. Comput. 23(2), 463–479 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Canuto, C., Simoncini, V., Verani, M.: On the decay of the inverse of matrices that are sum of kronecker products. Lin. Alg. Appl. 452, 21–39 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ċelik, C., Duman, M.: Crank–Nicolson method for the fractional diffusion equation with the Riesz fractional derivative. J. Comput Phys. 231(4), 1743–1750 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Dalton, S., Bell, N., Olson, L., Garland, M.: Cusp: generic parallel algorithms for sparse matrix and graph computations, version 0.5.0 (2014)
  12. 12.
    D’Ambra, P., Serafino, D.D., Filippone, S.: Mld2p4: a package of parallel algebraic multilevel domain decomposition preconditioners in fortran 95. ACM T. Math. Software 37(3), 30 (2010)MathSciNetGoogle Scholar
  13. 13.
    Demko, S., Moss, W.F., Smith, P.W.: Decay rates for inverses of band matrices. Math. Comput. 43(168), 491–499 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Deng, W.: Short memory principle and a predictor–corrector approach for fractional differential equations. J. Comput. Appl Math. 206(1), 174–188 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ding, H., Li, C., Chen, Y.: High–order algorithms for Riesz derivative and their applications (II). J. Comput. Phys. 293, 218–237 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Durastante, F.: Interpolant update of preconditioners for sequences of large linear systems. In: Mathematical Methods, Computational Techniques and Intelligent Systems (MAMECTIS ’15), vol. 41, pp. 40–47. WSEAS Press (2015)Google Scholar
  17. 17.
    Filippone, S., Colajanni, M.: Psblas: a library for parallel linear algebra computation on sparse matrices. ACM T. Math. Software 26(4), 527–550 (2000)CrossRefGoogle Scholar
  18. 18.
    Fornberg, B.: Classroom note: calculation of weights in finite difference formulas. SIAM Rev. 40(3), 685–691 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Gutknecht, M.H.: Variants of BICGSTAB for matrices with complex spectrum. SIAM J. Sci. Comput. 14(5), 1020–1033 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Jaffard, S.: Propriétés Des Matrices ≪Bien Localisées ≫ Près De Leur Diagonale Et Quelques Applications. In: Ann. I. H. Poincare-An, vol. 7, pp 461–476 (1990)Google Scholar
  21. 21.
    Krishtal, I., Strohmer, T., Wertz, T.: Localization of matrix factorizations. Found. Comput. Math. 15(4), 931–951 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Lampret, V.: Estimating the sequence of real binomial coefficients. J. Ineq. Pure and Appl. Math. 7(5)Google Scholar
  23. 23.
    Li, C., Chen, A., Ye, J.: Numerical approaches to fractional calculus and fractional ordinary differential equation. J. Comput. Phys. 230(9), 3352–3368 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Li, C., Zeng, F.: Numerical methods for fractional calculus, vol. 24. CRC Press (2015)Google Scholar
  25. 25.
    Lubich, C.: Discretized fractional calculus. SIAM J. Math. Anal. 17(3), 704–719 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Meerschaert, M.M., Tadjeran, C.: Finite difference approximations for fractional advection–dispersion flow equations. J. Comput. Appl. Math. 172(1), 65–77 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Ng, M.K., Pan, J.: Approximate inverse circulant–plus–diagonal preconditioners for Toeplitz–plus–diagonal matrices. SIAM J. Sci. Comput. 32(3), 1442–1464 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Ortigueira, M.D.: Riesz potential operators and inverses via fractional centred derivatives. Int. J. Math. Math. Sci (2006)Google Scholar
  29. 29.
    Pan, J., Ke, R., Ng, M.K., Sun, H.-W.: Preconditioning techniques for diagonal–times–Toeplitz matrices in fractional diffusion equations. SIAM J. Sci. Comput. 36(6), A2698–A2719 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Podlubny, I.: Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, vol. 198. Academic Press (1998)Google Scholar
  31. 31.
    Podlubny, I.: Matrix approach to discrete fractional calculus. Fract. Calc. Appl. Anal. 3(4), 359–386 (2000)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Podlubny, I.: Geometric and physical interpretation of fractional integration and fractional differentiation. Fract. Calc. Appl. Anal. 5(4), 367–386 (2002)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Podlubny, I., Chechkin, A., Skovranek, T., Chen, Y., Jara, B.M.V.: Matrix approach to discrete fractional calculus II: partial fractional differential equations. J. Comput. Phys. 228(8), 3137–3153 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Podlubny, I., Skovranek, T., Jara, B.M.V., Petras, I., Verbitsky, V., Chen, Y.: Matrix approach to discrete fractional calculus III: non-equidistant grids, variable step length and distributed orders. Philos. T. R. Soc. A 371(1990), 20120153 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Popolizio, M.: A matrix approach for partial differential equations with Riesz space fractional derivatives. Eur. Phys. J.-Spec. Top. 222(8), 1975–1985 (2013)CrossRefGoogle Scholar
  36. 36.
    Saad, Y.: Iterative methods for sparse linear systems. SIAM (2003)Google Scholar
  37. 37.
    Srivastava, H.M., Trujillo, J.J.: Theory and applications of fractional differential equations. Elsevier, Amsterdam (2006)zbMATHGoogle Scholar
  38. 38.
    Wendel, J.: Note on the gamma function. Am. Math. Mon. 55(9), 563–564 (1948)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Yang, Q., Liu, F., Turner, I.: Numerical methods for fractional partial differential equations with Riesz space fractional derivatives. Appl. Math. Model. 34(1), 200–218 (2010)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di Roma “Tor Vergata”RomaItaly
  2. 2.Dipartimento di Scienza e Alta TecnologiaUniversità dell’InsubriaComoItaly

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