Skip to main content
SpringerLink
Log in
Book cover

International Workshop on Eigenvalue Problems: Algorithms, Software and Applications in Petascale Computing

EPASA 2015: Eigenvalue Problems: Algorithms, Software and Applications in Petascale Computing pp 267–286Cite as

Fast Multipole Method as a Matrix-Free Hierarchical Low-Rank Approximation

Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE,volume 117)

Abstract

There has been a large increase in the amount of work on hierarchical low-rank approximation methods, where the interest is shared by multiple communities that previously did not intersect. This objective of this article is two-fold; to provide a thorough review of the recent advancements in this field from both analytical and algebraic perspectives, and to present a comparative benchmark of two highly optimized implementations of contrasting methods for some simple yet representative test cases. The first half of this paper has the form of a survey paper, to achieve the former objective. We categorize the recent advances in this field from the perspective of compute-memory tradeoff, which has not been considered in much detail in this area. Benchmark tests reveal that there is a large difference in the memory consumption and performance between the different methods.

This is a preview of subscription content, access via your institution.

References

  1. Ambikasaran, S., Darve, E.: An O(NlogN) fast direct solver for partial hierarchically semi-separable matrices. J. Sci. Comput. 57, 477–501 (2013)

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. Ambikasaran, S., Darve, E.: The inverse fast multipole method. arXiv:1407.1572v1 (2014)

    Google Scholar 

  3. Ambikasaran, S., Li, J.-Y., Kitanidis, P.K., Darve, E.: Large-scale stochastic linear inversion using hierarchical matrices. Comput. Geosci. 17(6), 913–927 (2013)

    CrossRef  MathSciNet  Google Scholar 

  4. Amestoy, P., Ashcraft, C., Boiteau, O., Buttari, A., L’Excellent, J.-Y., Weisbecker, C.: Improving multifrontal methods by means of block low-rank representations. SIAM J. Sci. Comput. 37(3), A1451–A1474 (2015)

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. Aminfar, A., Darve, E.: A fast, memory efficient and robust sparse preconditioner based on a multifrontal approach with applications to finite-element matrices. Int. J. Numer. Methods Eng. 107, 520–540 (2016)

    CrossRef  MathSciNet  MATH  Google Scholar 

  6. Aminfar, A., Ambikasaran, S., Darve, E.: A fast block low-rank dense solver with applications to finite-element matrices. J. Comput. Phys. 304, 170–188 (2016)

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. Anderson, C.R.: An implementation of the fast multipole method without multipoles. SIAM J. Sci. Stat. Comput. 13(4), 923–947 (1992)

    CrossRef  MathSciNet  MATH  Google Scholar 

  8. Appel, A.W.: An efficient program for many-body simulation. SIAM J. Sci. Stat. Comput. 6(1), 85–103 (1985)

    CrossRef  MathSciNet  Google Scholar 

  9. Barba, L.A., Yokota, R.: How will the fast multipole method fare in the exascale era? SIAM News 46(6), 1–3 (2013)

    Google Scholar 

  10. Barnes, J., Hut, P.: O(NlogN) force-calculation algorithm. Nature 324, 446–449 (1986)

    CrossRef  Google Scholar 

  11. Bebendorf, M.: Approximation of boundary element matrices. Numer. Math. 86, 565–589 (2000)

    CrossRef  MathSciNet  MATH  Google Scholar 

  12. Bebendorf, M.: Hierarchical Matrices. Lecture Notes in Computational Science and Engineering, vol. 63. Springer, Berlin (2008)

    Google Scholar 

  13. Bebendorf, M., Rjasanow, S.: Adaptive low-rank approximation of collocation matrices. Computing 70, 1–24 (2003)

    CrossRef  MathSciNet  MATH  Google Scholar 

  14. Bédorf, J., Gaburov, E., Fujii, M.S., Nitadori, K., Ishiyama, T., Portegies Zwart, S.: 24.77 Pflops on a gravitational tree-code to simulate the milky way galaxy with 18600 GPUs. In: Proceedings of the 2014 ACM/IEEE International Conference for High Performance Computing, Networking, Storage and Analysis, pp. 1–12 (2014)

    Google Scholar 

  15. Berman, C.L.: Grid-multipole calculations. SIAM J. Sci. Comput. 16(5), 1082–1091 (1995)

    CrossRef  MathSciNet  MATH  Google Scholar 

  16. Börm, S.: Construction of data-sparse h 2-matrices by hierarchical compression. SIAM J. Sci. Comput. 31(3), 1820–1839 (2009)

    CrossRef  MathSciNet  MATH  Google Scholar 

  17. Börm, S., Grasedyck, L.: Hybrid cross approximation of integral operators. Numer. Math. 101, 221–249 (2005)

    CrossRef  MathSciNet  MATH  Google Scholar 

  18. Börm, S., Grasedyck, L., Hackbusch, W.: Introduction to hierarchical matrices with applications. Eng. Anal. Bound. Elem. 27, 405–422 (2003)

    CrossRef  MATH  Google Scholar 

  19. Bremer, J.: A fast direct solver for the integral equations of scattering theory on planar curves with corners. J. Comput. Phys. 231, 1879–1899 (2012)

    CrossRef  MathSciNet  MATH  Google Scholar 

  20. Brunner, D., Junge, M., Rapp, P., Bebendorf, M., Gaul, L.: Comparison of the Fast Multipole Method with Hierarchical Matrices for the Helmholtz-BEM. Comput. Model. Eng. Sci. 58(2), 131–160 (2010)

    MathSciNet  MATH  Google Scholar 

  21. Burant, J.C., Strain, M.C., Scuseria, G.E., Frisch, M.J.: Analytic energy gradients for the Gaussian very fast multipole method (GvFMM). Chem. Phys. Lett. 248, 43–49 (1996)

    CrossRef  Google Scholar 

  22. Chaillat, S., Bonnet, M., Semblat, J.-F.: A multi-level fast multipole BEM for 3-D elastodynamics in the frequency domain. Comput. Methods Appl. Mech. Eng. 197, 4233–4249 (2008)

    CrossRef  MATH  Google Scholar 

  23. Chan, T.F.: On the existence and computation of LU-factorizations with small pivots. Math. Comput. 42(166), 535–547 (1984)

    MathSciNet  MATH  Google Scholar 

  24. Chan, T.F.: Rank revealing QR factorizations. Linear Algebra Appl. 88/89, 67–82 (1987)

    Google Scholar 

  25. Chandrasekaran, S., Ipsen, I.C.F.: On rank-revealing factorizations. SIAM J. Matrix Anal. Appl. 15(2), 592–622 (1994)

    CrossRef  MathSciNet  MATH  Google Scholar 

  26. Chandrasekaran, S., Dewilde, P., Gu, M., Lyons, W., Pals, T.: A fast solver for HSS representations via sparse matrices. SIAM J. Matrix Anal. Appl. 29(1), 67–81 (2006)

    CrossRef  MATH  Google Scholar 

  27. Chandrasekaran, S., Dewilde, P., Gu, M., Somasunderam, N.: On the numerical rank of the off-diagonal blocks of Schur complements of discretized elliptic PDEs. SIAM J. Matrix Anal. Appl. 31(5), 2261–2290 (2010)

    CrossRef  MathSciNet  MATH  Google Scholar 

  28. Cheng, H., Gimbutas, Z., Martinsson, P.G., Rokhlin, V.: On the compression of low rank matrices. SIAM J. Sci. Comput. 26(4), 1389–1404 (2005)

    CrossRef  MathSciNet  MATH  Google Scholar 

  29. Choi, C.H., Ruedenberg, K., Gordon, M.S.: New parallel optimal-parameter fast multipole method (OPFMM). J. Comput. Chem. 22(13), 1484–1501 (2001)

    CrossRef  Google Scholar 

  30. Corona, E., Martinsson, P.G., Zorin, D.: An O(N) direct solver for integral equations on the plane. Appl. Comput. Harmon. Anal. 38, 284–317 (2015)

    CrossRef  MathSciNet  MATH  Google Scholar 

  31. Coulaud, O., Fortin, P., Roman, J.: High performance BLAS formulation of the multipole-to-local operator in the fast multipole method. J. Comput. Phys. 227, 1836–1862 (2008)

    CrossRef  MathSciNet  MATH  Google Scholar 

  32. Dachsel, H.: Corrected article: “an error-controlled fast multipole method”. J. Chem. Phys. 132, 119901 (2010)

    CrossRef  Google Scholar 

  33. Darve, E., Havé, P.: A fast multipole method for Maxwell equations stable at all frequencies. Philos. Trans. R. Soc. Lond. A 362, 603–628 (2004)

    CrossRef  MathSciNet  MATH  Google Scholar 

  34. Darve, E., Cecka, C., Takahashi, T.: The fast multipole method on parallel clusters, multicore processors, and graphics processing units. C.R. Mec. 339, 185–193 (2011)

    Google Scholar 

  35. Dehnen, W.: A hierarchical O(N) force calculation algorithm. J. Comput. Phys. 179(1), 27–42 (2002)

    CrossRef  MathSciNet  MATH  Google Scholar 

  36. Dutt, A., Gu, M., Rokhlin, V.: Fast algorithms for polynomial interpolation, integration, and differntiation. SIAM J. Numer. Anal. 33(5), 1689–1711 (1996)

    CrossRef  MathSciNet  MATH  Google Scholar 

  37. Elliott, W.D., Board, J.A.: Fast Fourier transform accelerated fast multipole algorithm. SIAM J. Sci. Comput. 17(2), 398–415 (1996)

    CrossRef  MathSciNet  MATH  Google Scholar 

  38. Ethridge, F., Greengard, L.: A new fast-multipole accelerated Poisson solver in two dimensions. SIAM J. Sci. Comput. 23(3), 741–760 (2001)

    CrossRef  MathSciNet  MATH  Google Scholar 

  39. Fong, W., Darve, E.: The black-box fast multipole method. J. Comput. Phys. 228, 8712–8725 (2009)

    CrossRef  MathSciNet  MATH  Google Scholar 

  40. Fortin, P.: Multipole-to-local operator in the fast multipole method: Comparison of FFT, rotations and BLAS improvements. Technical Report RR-5752, Rapports de recherche, et theses de l’Inria (2005)

    Google Scholar 

  41. Gillman, A., Barnett, A., Martinsson, P.G.: A spectrally accurate direct solution technique for frequency-domain scattering problems with variable media. BIT Numer. Math. 55, 141–170 (2015)

    CrossRef  MathSciNet  MATH  Google Scholar 

  42. Gimbutas, Z., Greengard, L.: Fast multi-particle scattering: a hybrid solver for the Maxwell equations in microstructured materials. J. Comput. Phys. 232, 22–32 (2013)

    CrossRef  MathSciNet  MATH  Google Scholar 

  43. Gimbutas, Z., Rokhlin, V.: A generalized fast multipole method for nonoscillatory kernels. SIAM J. Sci. Comput. 24(3), 796–817 (2002)

    CrossRef  MathSciNet  MATH  Google Scholar 

  44. Goreinov, S.A., Tyrtyshnikov, E.E., Zamarashkin, N.L.: A theory of pseudoskeleton approximations. Linear Algebra Appl. 261(1–3), 1–21 (1997)

    CrossRef  MathSciNet  MATH  Google Scholar 

  45. Grasedyck, L., Hackbusch, W.: Construction and arithmetics of H-matrices. Computing 70, 295–334 (2003)

    CrossRef  MathSciNet  MATH  Google Scholar 

  46. Grasedyck, L., Kriemann, R., Le Borne, S.: Parallel black box H-LU preconditioning for elliptic boundary value problems. Comput. Vis. Sci. 11, 273–291 (2008)

    CrossRef  MathSciNet  Google Scholar 

  47. Grasedyck, L., Hackbusch, W., Kriemann, R.: Performance of H-LU preconditioning for sparse matrices. Comput. Methods Appl. Math. 8(4), 336–349 (2008)

    CrossRef  MathSciNet  MATH  Google Scholar 

  48. Grasedyck, L., Kriemann, R., Le Borne, S.: Domain decomposition based H-LU preconditioning. Numer. Math. 112, 565–600 (2009)

    CrossRef  MathSciNet  MATH  Google Scholar 

  49. Gray, A.G., Moore, A.W.: N-body problems in statistical learning. In: Leen, T.K., Dietterich, T.G., Tresp, V. (eds.) Advances in Neural Information Processing Systems, vol. 13, pp. 521—527. MIT Press, Cambridge (2001)

    Google Scholar 

  50. Greengard, L., Rokhlin, V.: A fast algorithm for particle simulations. J. Comput. Phys. 73(2), 325–348 (1987)

    CrossRef  MathSciNet  MATH  Google Scholar 

  51. Greengard, L., Rokhlin, V.: A new version of the fast multipole method for the Laplace equation in three dimensions. Acta Numer. 6, 229–269 (1997)

    CrossRef  MathSciNet  MATH  Google Scholar 

  52. Greengard, L., Gueyffier, D., Martinsson, P.G., Rokhlin, V.: Fast direct solvers for integral equations in complex three dimensional domains. Acta Numer. 18, 243–275 (2009)

    CrossRef  MathSciNet  MATH  Google Scholar 

  53. Gu, M., Eisenstat, S.C.: Efficient algorithms for computing a strong rank-revealing QR factorization. SIAM J. Sci. Comput. 17(4), 848–869 (1996)

    CrossRef  MathSciNet  MATH  Google Scholar 

  54. Gumerov, N.A., Duraiswami, R.: Fast radial basis function interpolation via preconditioned Krylov iteration. SIAM J. Sci. Comput. 29(5), 1876–1899 (2007)

    CrossRef  MathSciNet  MATH  Google Scholar 

  55. Hackbusch, W.: A sparse matrix arithmetic based on H-matrices, part I: Introduction to H-matrices. Computing 62, 89–108 (1999)

    CrossRef  MathSciNet  MATH  Google Scholar 

  56. Hackbusch, W., Nowak, Z.P.: On the fast matrix multiplication in the boundary element method by panel clustering. Numer. Math. 54, 463–491 (1989)

    CrossRef  MathSciNet  MATH  Google Scholar 

  57. Hackbusch, W., Khoromskij, B., Sauter, S.A.: On h 2-matrices. In: Bungartz, H., Hoppe, R., Zenger, C. (eds.) Lectures on Applied Mathematics. Springer, Berlin (2000)

    Google Scholar 

  58. Halko, N., Martinsson, P.G., Tropp, J.A.: Finding structure with randomness: probabilistic algorithms for constructing approximate matrix decompositions. SIAM Rev. 53(2), 217–288 (2011)

    CrossRef  MathSciNet  MATH  Google Scholar 

  59. Hao, S., Martinsson, P.G., Young, P.: An efficient and highly accurate solver for multi-body acoustic scattering problems involving rotationally symmetric scatterers. Comput. Math. Appl. 69, 304–318 (2015)

    CrossRef  MathSciNet  MATH  Google Scholar 

  60. Hénon, P., Saad, Y.: A parallel multistage ILU factorization based on a hierarchical graph decomposition. SIAM J. Sci. Comput. 28(6), 2266–2293 (2006)

    CrossRef  MathSciNet  MATH  Google Scholar 

  61. Hesford, A.J., Waag, R.C.: Reduced-rank approximations to the far-field transform in the gridded fast multipole method. J. Comput. Phys. 230, 3656–3667 (2011)

    CrossRef  MathSciNet  MATH  Google Scholar 

  62. Ho, K.L., Greengard, L.: A fast direct solver for structured linear systems by recursive skeletonization. SIAM J. Sci. Comput. 34(5), A2507–A2532 (2012)

    CrossRef  MathSciNet  MATH  Google Scholar 

  63. Ho, K.L., Ying, L.: Hierarchical interpolative factorization for elliptic operators: Integral equations. arXiv:1307.2666 (2015)

    Google Scholar 

  64. Hong, Y.P., Pan, C.T.: Rank-revealing QR factorizations and the singular value decomposition. Math. Comput. 58(197), 213–232 (1992)

    MathSciNet  MATH  Google Scholar 

  65. Hwang, T.-M., Lin, W.-W., Yang, E.K.: Rank revealing LU factorizations. Linear Algebra Appl. 175, 115–141 (1992)

    CrossRef  MathSciNet  MATH  Google Scholar 

  66. Hwang, T.-M., Lin, W.-W., Pierce, D.: Improved bound for rank revealing LU factorizations. Linear Algebra Appl. 261(1), 173–186 (1997)

    CrossRef  MathSciNet  MATH  Google Scholar 

  67. Ibeid, H., Yokota, R., Pestana, J., Keyes, D.: Fast multipole preconditioners for sparse matrices arising from elliptic equations. arXiv:1308.3339 (2016)

    Google Scholar 

  68. Izadi, M.: Hierarchical Matrix Techniques on Massively Parallel Computers. Ph.D. thesis, Universitat Leipzig (2012)

    Google Scholar 

  69. Kong, W.Y., Bremer, J., Rokhlin, V.: An adaptive fast direct solver for boundary integral equations in two dimensions. Appl. Comput. Harmon. Anal. 31, 346–369 (2011)

    CrossRef  MathSciNet  MATH  Google Scholar 

  70. Langston, H., Greengard, L., Zorin, D.: A free-space adaptive FMM-based PDE solver in three dimensions. Commun. Appl. Math. Comput. Sci. 6(1), 79–122 (2011)

    CrossRef  MathSciNet  MATH  Google Scholar 

  71. Le Borne, S.: Multilevel hierarchical matrices. SIAM J. Matrix Anal. Appl. 28(3), 871–889 (2006)

    CrossRef  MathSciNet  MATH  Google Scholar 

  72. Lee, D., Vuduc, R., Gray, A.G.: A distributed kernel summation framework for general-dimension machine learning. In: Proceedings of the 2012 SIAM International Conference on Data Mining (2012)

    Google Scholar 

  73. Lessel, K., Hartman, M., Chandrasekaran, S.: A fast memory efficient construction algorithm for hierarchically semi-separable representations. http://scg.ece.ucsb.edu/publications/MemoryEfficientHSS.pdf (2015)

  74. Li, J.-Y., Ambikasaran, S., Darve, E.F., Kitanidis, P.K.: A Kalman filter powered by h 2-matrices for quasi-continuous data assimilation problems. Water Resour. Res. 50, 3734–3749 (2014)

    CrossRef  Google Scholar 

  75. Liang, Z., Gimbutas, Z., Greengard, L., Huang, J., Jiang, S.: A fast multipole method for the Rotne-Prager-Yamakawa tensor and its applications. J. Comput. Phys. 234, 133–139 (2013)

    CrossRef  MathSciNet  MATH  Google Scholar 

  76. Liberty, E., Woolfe, F., Martinsson, P.G., Rokhlin, V., Tygert, M.: Randomized algorithms for the low-rank approximation of matrices. Proc. Natl. Acad. Sci. U.S.A. 104(51), 20167–20172 (2007)

    Google Scholar 

  77. Makino, J.: Yet another fast multipole method without multipoles – Pseudoparticle multipole method. J. Comput. Phys. 151(2), 910–920 (1999)

    CrossRef  MathSciNet  MATH  Google Scholar 

  78. Malhotra, D., Biros, G.: PVFMM: a parallel kernel independent FMM for particle and volume potentials. Commun. Comput. Phys. 18(3), 808–830 (2015)

    CrossRef  MathSciNet  MATH  Google Scholar 

  79. Malhotra, D., Gholami, A., Biros, G.: A volume integral equation stokes solver for problems with variable coefficients. In: Proceedings of the 2014 ACM/IEEE International Conference for High Performance Computing, Networking, Storage and Analysis, pp. 1–11 (2014)

    Google Scholar 

  80. March, W.B., Xiao, B., Biros, G.: ASKIT: approximate skeletonization kernel-independent treecode in high dimensions. SIAM J. Sci. Comput. 37(2), A1089–A1110 (2015)

    CrossRef  MathSciNet  MATH  Google Scholar 

  81. Martinsson, P.G.: The hierarchical Poincaré-Steklov (HPS) solver for elliptic PDEs: a tutorial. arXiv:1506.01308 (2015)

    Google Scholar 

  82. Martinsson, P.G., Rokhlin, V.: A fast direct solver for boundary integral equations in two dimensions. J. Comput. Phys. 205, 1–23 (2005)

    CrossRef  MathSciNet  MATH  Google Scholar 

  83. Miranian, L., Gu, M.: Strong rank revealing LU factorizations. Linear Algebra Appl. 367, 1–16 (2003)

    CrossRef  MathSciNet  MATH  Google Scholar 

  84. Ohno, Y., Yokota, R., Koyama, H., Morimoto, G., Hasegawa, A., Masumoto, G., Okimoto, N., Hirano, Y., Ibeid, H., Narumi, T., Taiji, M.: Petascale molecular dynamics simulation using the fast multipole method on k computer. Comput. Phys. Commun. 185, 2575–2585 (2014)

    CrossRef  Google Scholar 

  85. Oliveira, S., Yang F.: An algebraic approach for H-matrix preconditioners. Computing 80, 169–188 (2007)

    CrossRef  MathSciNet  MATH  Google Scholar 

  86. Pan, C.T.: On the existence and computation of rank-revealing LU factorizations. Linear Algebra Appl. 316, 199–222 (2000)

    CrossRef  MathSciNet  MATH  Google Scholar 

  87. Petersen, H.G., Soelvason, D., Perram, J.W., Smith, E.R.: The very fast multipole method. J. Chem. Phys. 101(10), 8870–8876 (1994)

    CrossRef  Google Scholar 

  88. Rahimian, A., Lashuk, I., Veerapaneni, K., Chandramowlishwaran, A., Malhotra, D., Moon, L., Sampath, R., Shringarpure, A., Vetter, J., Vuduc, R., Zorin, D., Biros, G.: Petascale direct numerical simulation of blood flow on 200k cores and heterogeneous architectures. In: Proceedings of the 2010 ACM/IEEE International Conference for High Performance Computing, Networking, Storage and Analysis, SC ’10 (2010)

    Google Scholar 

  89. Rouet, F.-H., Li, X.-S., Ghysels, P., Napov, A.: A distributed-memory package for dense hierarchically semi-separable matrix computations using randomization. arXiv:1503.05464 (2015)

    Google Scholar 

  90. Shao, Y., White, C.A., Head-Gordon, M.: Efficient evaluation of the Coulomb force in density-functional theory calculations. J. Chem. Phys. 114(15), 6572–6577 (2001)

    CrossRef  Google Scholar 

  91. Takahashi, T., Cecka, C., Fong, W., Darve, E.: Optimizing the multipole-to-local operator in the fast multipole method for graphical processing units. Int. J. Numer. Methods Eng. 89, 105–133 (2012)

    CrossRef  MATH  Google Scholar 

  92. Verde, A., Ghassemi, A.: Fast multipole displacement discontinuity method (FM-DDM) for geomechanics reservoir simulations. Int. J. Numer. Anal. Methods Geomech. 39(18), 1953–1974 (2015)

    CrossRef  Google Scholar 

  93. Wang, Y., Wang, Q., Deng, X., Xia, Z., Yan, J., Xu, H.: Graphics processing unit (GPU) accelerated fast multipole BEM with level-skip M2L for 3D elasticity problems. Adv. Eng. Softw. 82, 105–118 (2015)

    CrossRef  Google Scholar 

  94. White, C.A., Head-Gordon, M.: Rotating around the quartic angular momentum barrier in fast multipole method calculations. J. Chem. Phys. 105(12), 5061–5067 (1996)

    CrossRef  Google Scholar 

  95. Wilkes, D.R., Duncan, A.J.: A low frequency elastodynamic fast multipole boundary element method in three dimensions. Comput. Mech. 56, 829–848 (2015)

    CrossRef  MathSciNet  MATH  Google Scholar 

  96. Willis, D., Peraire, J., White, J.: FastAero – a precorrected FFT-fast multipole tree steady and unsteady potential flow solver. http://hdl.handle.net/1721.1/7378 (2005)

  97. Wolf, W.R., Lele, S.K.: Aeroacoustic integrals accelerated by fast multipole method. AIAA J. 49(7), 1466–1477 (2011)

    CrossRef  Google Scholar 

  98. Xia, J.: Randomized sparse direct solvers. SIAM J. Matrix Anal. Appl. 34(1), 197–227 (2013)

    CrossRef  MathSciNet  MATH  Google Scholar 

  99. Xia, J.: O(N) complexity randomized 3D direct solver with HSS2D structure. In: Proceedings of the Project Review, Geo-Mathematical Imaging Group, Purdue University, pp. 317–325 (2014)

    Google Scholar 

  100. Xia, J., Chandrasekaran, S., Gu, M., Li, X.S.: Superfast multifrontal method for large structured linear systems of equations. SIAM J. Matrix Anal. Appl. 31(3), 1382–1411 (2009)

    CrossRef  MathSciNet  MATH  Google Scholar 

  101. Xia, J., Chandrasekaran, S., Gu, M., Li, X.S.: Fast algorithms for hierarchically semiseperable matrices. Numer. Linear Algebra Appl. 17, 953–976 (2010)

    CrossRef  MathSciNet  MATH  Google Scholar 

  102. Yarvin, N., Rokhlin, V.: An improved fast multipole algorithm for potential fields on the line. SIAM J. Numer. Anal. 36(2), 629–666 (1999)

    CrossRef  MathSciNet  MATH  Google Scholar 

  103. Ying, L., Biros, G., Zorin, D.: A kernel-independent adaptive fast multipole algorithm in two and three dimensions. J. Comput. Phys. 196(2), 591–626 (2004)

    CrossRef  MathSciNet  MATH  Google Scholar 

  104. Ying, L., Biros, G., Zorin, D.: A high-order 3D boundary integral equation solver for elliptic PDEs in smooth domains. J. Comput. Phys. 219, 247–275 (2006)

    CrossRef  MathSciNet  MATH  Google Scholar 

  105. Yokota, R., Bardhan, J.P., Knepley, M.G., Barba, L.A., Hamada, T.: Biomolecular electrostatics using a fast multipole BEM on up to 512 GPUs and a billion unknowns. Comput. Phys. Commun. 182, 1272–1283 (2011)

    CrossRef  MATH  Google Scholar 

  106. Yokota, R., Narumi, T., Yasuoka, K., Barba, L.A.: Petascale turbulence simulation using a highly parallel fast multipole method on GPUs. Comput. Phys. Commun. 184, 445–455 (2013)

    CrossRef  MathSciNet  Google Scholar 

  107. Yunis, E., Yokota, R., Ahmadia, A.: Scalable force directed graph layout algorithms using fast multipole methods. In: The 11th International Symposium on Parallel and Distributed Computing, Munich, June 2012

    Google Scholar 

  108. Zhao, Z., Kovvali, N., Lin, W., Ahn, C.-H., Couchman, L., Carin, L.: Volumetric fast multipole method for modeling Schrödinger’s equation. J. Comput. Phys. 224, 941–955 (2007)

    CrossRef  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We thank François-Henry Rouet, Pieter Ghysels, and Xiaoye, S. Li for providing the STRUMPACK interface for our comparisons between FMM and HSS. This work was supported by JSPS KAKENHI Grant-in-Aid for Research Activity Start-up Grant Number 15H06196. This publication was based on work supported in part by Award No KUK-C1-013-04, made by King Abdullah University of Science and Technology (KAUST). This work used the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation grant number OCI-1053575.

Author information

Authors and Affiliations

  1. Tokyo Institute of Technology, 2-12-1 O-okayama Meguro-ku, Tokyo, Japan

    Rio Yokota

  2. King Abdullah University of Science and Technology, 4700 KAUST, Thuwal, Saudi Arabia

    Huda Ibeid & David Keyes

Authors
  1. Rio Yokota
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Huda Ibeid
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. David Keyes
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rio Yokota .

Editor information

Editors and Affiliations

  1. Department of Computer Science, University of Tsukuba, Tsukuba, Ibaraki, Japan

    Prof. Tetsuya Sakurai

  2. Department of Electrical Engineering and Computer Science, Nagoya University, Nagoya, Aichi, Japan

    Prof. Shao-Liang Zhang

  3. RIKEN Advanced Institute for Computational Science, Kobe, Hyogo, Japan

    Dr. Toshiyuki Imamura

  4. Department of Communication Engineering and Informatics, University of Electro-Communications, Graduate School of Informatics and Engineering, Tokyo, Japan

    Prof. Yusaku Yamamoto

  5. Department of Particle Physics, University of Tsukuba, Tsukuba, Ibaraki, Japan

    Prof. Yoshinobu Kuramashi

  6. Department of Applied Mathematics and Physics, Tottori University, Tottori, Japan

    Prof. Takeo Hoshi

Rights and permissions

Reprints and Permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Cite this paper

Yokota, R., Ibeid, H., Keyes, D. (2017). Fast Multipole Method as a Matrix-Free Hierarchical Low-Rank Approximation. In: Sakurai, T., Zhang, SL., Imamura, T., Yamamoto, Y., Kuramashi, Y., Hoshi, T. (eds) Eigenvalue Problems: Algorithms, Software and Applications in Petascale Computing. EPASA 2015. Lecture Notes in Computational Science and Engineering, vol 117. Springer, Cham. https://doi.org/10.1007/978-3-319-62426-6_17

Download citation

search

Navigation