\({\mathscr{H}}\)-matrix approximability of inverses of discretizations of the fractional Laplacian

  • Michael KarkulikEmail author
  • Jens Markus Melenk


The integral version of the fractional Laplacian on a bounded domain is discretized by a Galerkin approximation based on piecewise linear functions on a quasiuniform mesh. We show that the inverse of the associated stiffness matrix can be approximated by blockwise low-rank matrices at an exponential rate in the block rank.


Hierarchical matrices Fractional Laplacian 

Mathematics Subject Classification (2010)

65N30 65F05 65F30 65F50 


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Funding information

MK was supported by Conicyt Chile through project FONDECYT 1170672. JMM was supported by the Austrian Science Fund (FWF) project F 65.


  1. 1.
    Acosta, G., Bersetche, F.M., Borthagaray, J.P.: A short FE implementation for a 2d homogeneous Dirichlet problem of a fractional Laplacian. Comput. Math Appl. 74(4), 784–816 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Acosta, G., Borthagaray, J.P.: A fractional Laplace equation: regularity of solutions and finite element approximations. SIAM J. Numer. Anal. 55(2), 472–495 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Adams, R.A., Fournier, J.J.F.: Sobolev Spaces, Volume 140 of Pure and Applied Mathematics (Amsterdam), 2nd edn. Elsevier/Academic Press, Amsterdam (2003)Google Scholar
  4. 4.
    Ainsworth, M., Glusa, C.: Aspects of an adaptive finite element method for the fractional Laplacian: a priori and a posteriori error estimates, efficient implementation and multigrid solver. Comput. Methods Appl. Mech. Engrg. 327, 4–35 (2017)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bebendorf, M.: Approximation of boundary element matrices. Numer. Math. 86(4), 565–589 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Bebendorf, M.: Hierarchical LU decomposition-based preconditioners for BEM. Computing 74(3), 225–247 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Bebendorf, M.: Why finite element discretizations can be factored by triangular hierarchical matrices. SIAM J. Numer. Anal. 45(4), 1472–1494 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Bonito, A., Lei, W., Pasciak, J.E.: On sinc quadrature approximations of fractional powers of regularly accretive operators. ArXiv e-prints (2017)Google Scholar
  9. 9.
    Bonito, A., Pasciak, J. : Numerical approximation of fractional powers of elliptic operators. Math. Comp. 84(295), 2083–2110 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Börm, S.: Approximation of solution operators of elliptic partial differential equations by \({\mathscr{H}}^{2}\)-matrices. Numer. Math. 115(2), 165–193 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Börm, S.: Efficient Numerical Methods for Non-Local Operators, Volume 14 of EMS Tracts in Mathematics. European Mathematical Society (EMS), Zürich (2010)CrossRefGoogle Scholar
  12. 12.
    Börm, S., Grasedyck, L.: Hybrid cross approximation of integral operators. Numer. Math. 101(2), 221–249 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Cabré, X., Sire, Y.: Nonlinear equations for fractional Laplacians, I Regularity, maximum principles, and Hamiltonian estimates. Ann. Inst. H. Poincaré, Anal. Non Linéaire 31(1), 23–53 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Cabré, X., Tan, J.: Positive solutions of nonlinear problems involving the square root of the Laplacian. Adv. Math. 224(5), 2052–2093 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Comm. Partial Diff. Equ. 32(7-9), 1245–1260 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Caffarelli, L., Stinga, P.: Fractional elliptic equations, Caccioppoli estimates and regularity. Ann. Inst. H. Poincaré, Anal. Non Linéaire 33(3), 767–807 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Capella, A., Dávila, J., Dupaigne, L., Sire, Y.: Regularity of radial extremal solutions for some non-local semilinear equations. Comm Partial Diff. Equ. 36(8), 1353–1384 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Corona, E., Martinsson, P.-G., Zorin, D.: An O(N) direct solver for integral equations on the plane. Appl. Comput. Harmon. Anal. 38(2), 284–317 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Deny, J., Lions, J.L.: Les espaces du type de Beppo Levi. Ann. Fourier, Grenoble 5, 305–370 (1955). 1953–54MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Dölz, J., Harbrecht, H., Schwab, C.: Covariance regularity and \({\mathscr{H}}\)-matrix approximation for rough random fields. Numer. Math. 135(4), 1045–1071 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Duan, R., Lazarov, R., P.J.: Numerical approximation of fractional powers of elliptic operators. Technical report, arXiv:1803.10055 (2018)
  22. 22.
    Faustmann, M., Melenk, J.M., Praetorius, D.: \({\mathscr{H}}\)-matrix approximability of the inverses of FEM matrices. Numer. Math. 131(4), 615–642 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Faustmann, M., Melenk, J.M., Praetorius, D.: Existence of \({\mathscr{H}}\)-matrix approximants to the inverses of BEM matrices: the simple-layer operator. Math. Comp. 85(297), 119–152 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Faustmann, M., Melenk, J.M., Praetorius, D.: Existence of \({\mathscr{H}}\)-matrix approximants to the inverse of BEM matrices: the hyper-singular integral operator. IMA J. Numer. Anal. 37(3), 1211–1244 (2017)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Gagliardo, E.: Caratterizzazioni delle tracce sulla frontiera relative ad alcune classi di funzioni in n variabili. Rend. Sem. Mat. Univ. Padova 27, 284–305 (1957)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Gavrilyuk, I.P., Hackbusch, W., Khoromskij, B.N.: Hierarchical tensor-product approximation to the inverse and related operators for high-dimensional elliptic problems. Computing 74(2), 131–157 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Gillman, A., Martinsson, P.G.: A direct solver with O(N) complexity for variable coefficient elliptic PDEs discretized via a high-order composite spectral collocation method. SIAM J. Sci. Comput. 36(4), A2023–A2046 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Gol’dshtein, V., Ukhlov, A.: Weighted Sobolev spaces and embedding theorems. Trans. Amer. Math. Soc. 361(7), 3829–3850 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Grasedyck, L.: Theorie und Anwendungen Hierarchischer Matrizen. Doctoral thesis Kiel (2001)Google Scholar
  30. 30.
    Grasedyck, L.: Adaptive recompression of \({\mathscr{H}}\)-matrices for BEM. Computing 74(3), 205–223 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Grasedyck, L., Hackbusch, W.: Construction and arithmetics of \({\mathscr{H}}\)-matrices. Computing 70(4), 295–334 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Grasedyck, L., Hackbusch, W., Kriemann, R.: Performance of \({\mathscr{H}}\)-LU preconditioning for sparse matrices. Comput. Methods Appl. Math. 8(4), 336–349 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Grasedyck, L., Kriemann, R., Le Borne, S.: Parallel black box \({\mathscr{H}}\)-LU preconditioning for elliptic boundary value problems. Comput. Vis. Sci. 11(4-6), 273–291 (2008)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Greengard, L., Gueyffier, D., Martinsson, P., Rokhlin, V.: Fast direct solvers for integral equations in complex three-dimensional domains. Acta Numer. 18, 243–275 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Hackbusch, W.: A sparse matrix arithmetic based on \({\mathscr{H}}\)-matrices. Introduction to \({\mathscr{H}}\)-matrices. Computing 62(2), 89–108 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Hackbusch, W.: Hierarchical Matrices: Algorithms and Analysis, Volume 49 of Springer Series in Computational Mathematics. Springer, Heidelberg (2015)CrossRefGoogle Scholar
  37. 37.
    Hackbusch, W., Khoromskij, B., Sauter, S.: On \({\mathscr{H}}^{2}\)-matrices. Lect. Appl. Math., 9–29 (2000)Google Scholar
  38. 38.
    Hackbusch, W., Khoromskij, B.N.: A sparse \({\mathscr{H}}\)-matrix arithmetic: general complexity estimates. J. Comput. Appl. Math. 125(1-2), 479–501 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Hackbusch, W., Khoromskij, B.N.: A sparse \({\mathscr{H}}\)-matrix arithmetic. II. Application to multi-dimensional problems. Computing 64(1), 21–47 (2000)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Hale, N., Higham, N.J., Trefethen, L.N.: Computing A \(^{{\alpha }},\log (A)\), and related matrix functions by contour integrals. SIAM J. Numer. Anal. 46(5), 2505–2523 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Harizanov, S., Lazarov, R., Margenov, S., Marinov, P., Vutov, Y.: Optimal solvers for linear systems with fractional powers of sparse SPD matrices. Numer Linear Algebra Appl. 25(5), e2167,24 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Heidel, G., Khoromskaia, V., Khoromskij, B., Schulz, V.: Tensor approach to optimal control problems with fractional d-dimensional elliptic operator in constraints. Technical report, arXiv:1809.01971 (2018)
  43. 43.
    Ho, K., Greengard, L.: A fast direct solver for structured linear systems by recursive skeletonization. SIAM J. Sci Comput. 34(5), A2507–A2532 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Ho, K.L., Ying, L.: Hierarchical interpolative factorization for elliptic operators: differential equations. Comm. Pure Appl. Math. 69(8), 1415–1451 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Ho, K.L., Ying, L.: Hierarchical interpolative factorization for elliptic operators: integral equations. Comm. Pure Appl. Math. 69(7), 1314–1353 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Kufner, A.: Weighted Sobolev Spaces. A Wiley-Interscience Publication. Wiley, New York (1985). Translated from the CzechGoogle Scholar
  47. 47.
    Kwaśnicki, M.: Ten equivalent definitions of the fractional Laplace operator. Fract. Calc. Appl. Anal. 20(1), 7–51 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Le Borne, S., Grasedyck, L.: \({\mathscr{H}}\)-matrix preconditioners in convection-dominated problems. SIAM J. Matrix Anal. Appl. 27 (4), 1172–1183 (2006). (electronic)MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    Li, I., Mao, Z., Song, F., Wang, H., Karniadakis, G.: A fast solver for spectral element approximation applied to fractional differential equations using hierarchical matrix approximation. Technical report (2018)
  50. 50.
    Li, S., Gu, M., Wu, C., Xia, J.: New efficient and robust HSS Cholesky factorization of SPD matrices. SIAM J. Matrix Anal. Appl. 33(3), 886–904 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    Lischke, A., Pang, G., Gulian, M., Song, F., Glusa, C., Zheng, X., Mao, Z., Cai, W., Meers chaert, M., Ainsworth, M., Karniadakis, G.: What is the fractional Laplacian? Technical report, arxiv:1801.09767, 01 (2018)
  52. 52.
    Martinsson, P.: A fast direct solver for a class of elliptic partial differential equations. J. Sci. Comput. 38(3), 316–330 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    Martinsson, P., Rokhlin, V.: A fast direct solver for boundary integral equations in two dimensions. J. Comput. Phys. 205(1), 1–23 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  54. 54.
    Muckenhoupt, B.: Hardy’s inequality with weights. Studia Math. 44, 31–38 (1972). Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, IMathSciNetzbMATHCrossRefGoogle Scholar
  55. 55.
    Nochetto, R.H., Otárola, E., Salgado, A.J.: Piecewise polynomial interpolation in Muckenhoupt weighted Sobolev spaces and applications. Numer. Math. 132(1), 85–130 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  56. 56.
    Schmitz, P., Ying, L.: A fast direct solver for elliptic problems on general meshes in 2D. J. Comput. Phys. 231(4), 1314–1338 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  57. 57.
    Scott, L.R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp. 54(190), 483–493 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  58. 58.
    Stinga, P., Torrea, J.: Extension problem and Harnack’s inequality for some fractional operators. Comm. Partial Diff. Equ. 35(11), 2092–2122 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  59. 59.
    Xia, J.: Efficient structured multifrontal factorization for general large sparse matrices. SIAM J. Sci. Comput. 35(2), A832–A860 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  60. 60.
    Xia, J., Chandrasekaran, S., Gu, M., Li, X.: Superfast multifrontal method for large structured linear systems of equations. SIAM J Matrix Anal. Appl. 31(3), 1382–1411 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  61. 61.
    Zhao, X., Hu, X., Cai, W., Karniadakis, G.E : Adaptive finite element method for fractional differential equations using hierarchical matrices. Comput. Methods Appl. Mech Engrg. 325, 56–76 (2017)MathSciNetCrossRefGoogle Scholar
  62. 62.
    Zygmund, A.: Trigonometric Series. Cambridge Mathematical Library, 3rd edn., vol. I, II. Cambridge University Press, Cambridge (2002). With a foreword by Robert A. FeffermanGoogle Scholar

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Authors and Affiliations

  1. 1.Departamento de MatemáticaUniversidad Técnica Federico Santa MaríaValparaísoChile
  2. 2.Institut für Analysis und Scientific ComputingTechnische Universität WienViennaAustria

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