An O(N log N) Fast Direct Solver for Partial Hierarchically Semi-Separable Matrices: With Application to Radial Basis Function Interpolation

Journal of Scientific Computing

Volumn 57, Issue 3, 2013, Pages 477-501

  Ambikasaran, Sivaram ^a   Darve, Eric ^a  

^a Stanford University   (United States)

Author keywords

Fast direct solver; Hierarchical matrix; Numerical linear algebra; Partial hierarchically semi separable representation; Radial basis function

Indexed keywords

DIRECT SOLVERS; HIERARCHICAL MATRICES; NUMERICAL LINEAR ALGEBRA; PARTIAL HIERARCHICALLY SEMI-SEPARABLE REPRESENTATION; RADIAL BASIS FUNCTIONS;

ALGORITHMS; IMAGE SEGMENTATION; INTERPOLATION; ITERATIVE METHODS; LINEAR ALGEBRA; NUMERICAL METHODS;

RADIAL BASIS FUNCTION NETWORKS;

EID: 84889083830     PISSN: 08857474     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10915-013-9714-z     Document Type: Article

References (65)

1. Andrianakis, I.; Challenor, P.: The effect of the nugget on gaussian process emulators of computer models. Comput. Stat. Data Anal. 56(12), 4215-4228 (2012)
2. Arnoldi, W.: The principle of minimized iterations in the solution of the matrix eigenvalue problem. Quart. Appl. Math. 9(1), 17-29 (1951)
3. Barnes, J.; Hut, P.: A hierarchical O(N log N) force-calculation algorithm. Nature 324(4), 446-449 (1986)
4. Baxter, B.: The interpolation theory of radial basis functions. ArXiv, preprint arXiv:1006.2443 (2010)
5. Beatson, R.; Cherrie, J.; Mouat, C.: Fast fitting of radial basis functions: methods based on preconditioned GMRES iteration. Adv. Comput. Math. 11(2), 253-270 (1999)
6. Beatson, R.; Greengard, L.: A short course on fast multipole methods. Wavelets, Multilevel Methods Elliptic PDEs pp. 1-37 (1997)
7. Beatson, R.; Newsam, G.: Fast evaluation of radial basis functions: I. Comput. Math. Appl. 24(12), 7-19 (1992)
8. Billings, S.; Beatson, R.; Newsam, G.: Interpolation of geophysical data using continuous global surfaces. Geophysics 67(6), 1810-1822 (2002)
9. Börm, S.; Grasedyck, L.; Hackbusch, W.: Hierarchical matrices. In: Lecture notes 21 (2005)
10. Buhmann, M.: Radial Basis Functions: Theory and Implementations, vol. 12. Cambridge University Press, Cambridge (2003)
11. Chandrasekaran, S.; Dewilde, P.; Gu, M.; Pals, T.; Sun, X.; van der Veen, A.; White, D.: Some fast algorithms for sequentially semiseparable representations. SIAM J. Matrix Anal. Appl. 27(2), 341-364 (2006)
12. Chandrasekaran, S.; Gu, M.; Pals, T.: A fast ULV decomposition solver for hierarchically semiseparable representations. SIAM J. Matrix Anal. Appl. 28(3), 603-622 (2006)
13. Chen, K.: An analysis of sparse approximate inverse preconditioners for boundary integral equations. SIAM J. Matrix Anal. Appl. 22, 1058 (2001)
14. Cheng, H.; Gimbutas, Z.; Martinsson, P.; Rokhlin, V.: On the compression of low rank matrices. SIAM J. Sci. Comput. 26(4), 1389-1404 (2005)
15. Cheng, H.; Greengard, L.; Rokhlin, V.: A fast adaptive multipole algorithm in three dimensions. J. Comput. Phys. 155(2), 468-498 (1999)
16. Coifman, R.; Rokhlin, V.; Wandzura, S.: The fast multipole method for the wave equation: a pedestrian prescription. IEEE Antennas Propag. Mag. 35(3), 7-12 (1993)
17. Darve, E.: The fast multipole method: numerical implementation. J. Comput. Phys. 160(1), 195-240 (2000)
18. Darve, E.: The fast multipole method. I: error analysis and asymptotic complexity. SIAM J. Numer. Anal. 38(1), 98-128 (2000)
19. Davis, G.; Morris, M.: Six factors which affect the condition number of matrices associated with kriging. Math. Geol. 29(5), 669-683 (1997)
20. De Boer, A.; Van der Schoot, M.; Bijl, H.: Mesh deformation based on radial basis function interpolation. Comput. Struct. 85(11-14), 784-795 (2007)
21. Dietrich, C.; Newsam, G.: Efficient generation of conditional simulations by Chebyshev matrix polynomial approximations to the symmetric square root of the covariance matrix. Math. Geol. 27(2), 207-228 (1995)
22. Fong, W.; Darve, E.: The black-box fast multipole method. J. Comput. Phys. 228(23), 8712-8725 (2009)
23. Freund, R.: A transpose-free quasi-minimal residual algorithm for non-Hermitian linear systems. SIAM J. Sci. Comput. 14, 470-482 (1993)
24. Freund, R.; Nachtigal, N.: QMR: a quasi-minimal residual method for non-Hermitian linear systems. Numerische Mathematik 60(1), 315-339 (1991)
25. Frieze, A.; Kannan, R.; Vempala, S.: Fast Monte-Carlo algorithms for finding low-rank approximations. J. ACM (JACM) 51(6), 1025-1041 (2004)
26. Gillman, A.; Young, P.; Martinsson, P.: A direct solver with O(N) complexity for integral equations on one-dimensional domains. ArXiv, preprint arXiv:1105.5372 (2011)
27. Golub, G.; Van Loan, C.: Matrix Computations, vol. 3. Johns Hopkins University Press, Baltimore (1996)
28. Goreinov, S.; Tyrtyshnikov, E.; Zamarashkin, N.: A theory of pseudoskeleton approximations. Linear Algebra Appl. 261(1-3), 1-21 (1997)
29. Grasedyck, L.; Hackbusch, W.: Construction and arithmetics of H-matrices. Computing 70(4), 295-334 (2003)
30. Greengard, L.; Gueyffier, D.; Martinsson, P.; Rokhlin, V.: Fast direct solvers for integral equations in complex three-dimensional domains. Acta Numerica 18(1), 243-275 (2009)
31. Greengard, L.; Rokhlin, V.: A fast algorithm for particle simulations. J. Comput. Phys. 73(2), 325-348 (1987)
32. Greengard, L.; Rokhlin, V.: A new version of the fast multipole method for the Laplace equation in three dimensions. Acta Numerica 6(1), 229-269 (1997)
33. Gu, M.; Eisenstat, S.: Efficient algorithms for computing a strong rank-revealing QR factorization. SIAM J. Sci. Comput. 17(4), 848-869 (1996)
34. Guennebaud, G.; Jacob, B.; et al.: Eigen v3. http://eigen.tuxfamily.org (2010)
35. Gumerov, N.; Duraiswami, R.: Fast radial basis function interpolation via preconditioned Krylov iteration. SIAM J. Sci. Comput. 29(5), 1876-1899 (2007)
36. Hackbusch, W.: A sparse matrix arithmetic based on H-matrices. Part I: introduction to H-matrices. Computing 62(2), 89-108 (1999)
37. Hackbusch, W.; Börm, S.: Data-sparse approximation by adaptive H2-matrices. Computing 69(1), 1-35 (2002)
38. Hackbusch, W.; Khoromskij, B.: A sparse H-matrix arithmetic. Computing 64(1), 21-47 (2000)
39. Hackbusch, W.; Nowak, Z.: On the fast matrix multiplication in the boundary element method by panel clustering. Numerische Mathematik 54(4), 463-491 (1989)
40. Hager, W.: Updating the inverse of a matrix. SIAM Rev. 31(2), 221-239 (1989)
41. Hestenes, M.; Stiefel, E.: Methods of conjugate gradients for solving linear systems. J. Res. Natl. Bur. Stand. 49(6), 409-436 (1952)
42. Kong, W.; Bremer, J.; Rokhlin, V.: An adaptive fast direct solver for boundary integral equations in two dimensions. Appl. Comput. Harm. Anal. 31(3), 346-369 (2011)
43. Liberty, E.; Woolfe, F.; Martinsson, P.; Rokhlin, V.; Tygert, M.: Randomized algorithms for the low-rank approximation of matrices. Proc. Natl. Acad. Sci. 104, 20167-20172 (2007)
44. Martinsson, P.: A fast direct solver for a class of elliptic partial differential equations. J. Sci. Comput. 38(3), 316-330 (2009)
45. Martinsson, P.; Rokhlin, V.: A fast direct solver for boundary integral equations in two dimensions. J. Comput. Phys. 205(1), 1-23 (2005)
46. Messner, M.; Schanz, M.; Darve, E.: Fast directional multilevel summation for oscillatory kernels based on Chebyshev interpolation. J. Comp. Phys. 231(4), 1175-1196 (2012)
47. Miranian, L.; Gu, M.: Strong rank revealing LU factorizations. Linear Algebra Appl. 367, 1-16 (2003)
48. Nishimura, N.: Fast multipole accelerated boundary integral equation methods. Appl. Mech. Rev. 55(4), 299-324 (2002)
49. O'Dowd, R.: Conditioning of coefficient matrices of ordinary kriging. Math. Geol. 23(5), 721-739 (1991)
50. Paige, C.; Saunders, M.: Solution of sparse indefinite systems of linear equations. SIAM J. Numer. Anal. 12(4), 617-629 (1975)
51. Pan, C.T.: On the existence and computation of rank-revealing LU factorizations. Linear Algebra Appl. 316(1), 199-222 (2000)
52. Rjasanow, S.: Adaptive cross approximation of dense matrices. In: IABEM 2002, International Association for Boundary Element Methods (2002)
53. Saad, Y.; Schultz, M.: GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7(3), 856-869 (1986)
54. Schaback, R.: Creating surfaces from scattered data using radial basis functions. Math. Methods Curves Surf. 477-496 (1995)
55. Schmitz, P.; Ying, L.: A fast direct solver for elliptic problems on general meshes in 2D. J. Comput. Phys. 231(4), 1314-1338 (2012)
56. Schmitz, P.; Ying, L.: A fast direct solver for elliptic problems on Cartesian meshes in 3D (in review) (2012)
57. Vandebril, R.; Barel, M.; Golub, G.; Mastronardi, N.: A bibliography on semiseparable matrices. Calcolo 42(3), 249-270 (2005)
58. Vavasis, S.: Preconditioning for boundary integral equations. SIAM J. Matrix Anal. Appl. 13, 905-925 (1992)
59. Van der Vorst, H.: Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of non-symmetric linear systems. SIAM J. Sci. Stat. Comput. 13(2), 631-644 (1992)
60. Wang, J.; Liu, G.: A point interpolation meshless method based on radial basis functions. Int. J. Numer. Methods Eng. 54(11), 1623-1648 (2002)
61. Woodbury, M.A.: Inverting modified matrices. Statistical Research Group, Memo. Rep. no. 42, Princeton University, Princeton (1950)
62. Woolfe, F.; Liberty, E.; Rokhlin, V.; Tygert, M.: A fast randomized algorithm for the approximation of matrices. Appl. Comput. Harmon. Anal. 25(3), 335-366 (2008)
63. Wu, Z.; Schaback, R.: Local error estimates for radial basis function interpolation of scattered data. IMA J. Numer. Anal. 13(1), 13-27 (1993)
64. Xia, J.; Chandrasekaran, S.; Gu, M.; Li, X.: Fast algorithms for hierarchically semiseparable matrices. Numer. Linear Algebra Appl. 17(6), 953-976 (2010)
65. 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)