메뉴 건너뛰기




Volumn 33, Issue 2, 2012, Pages 388-410

On the complexity of some hierarchical structured matrix algorithms

Author keywords

Hierarchically semiseparable (HSS) matrix; HSS construction and ULV factorization; Rank structure; Recompression; Relaxation of rank requirement

Indexed keywords

COMPLEXITY ANALYSIS; EIGEN-PROBLEMS; FAST SOLUTIONS; GRAPH TECHNIQUE; MAXIMUM RANK; NUMERICAL EXAMPLE; RANK STRUCTURE; RECOMPRESSION; RELAXATION OF RANK REQUIREMENT; SOLUTION ALGORITHMS; STRUCTURED MATRIXES; TOEPLITZ;

EID: 84865452975     PISSN: 08954798     EISSN: 10957162     Source Type: Journal    
DOI: 10.1137/110827788     Document Type: Article
Times cited : (45)

References (43)
  • 1
    • 37249058366 scopus 로고    scopus 로고
    • Why finite element discretizations can be factored by triangular hierarchical matrices
    • M. Bebendorf, Why finite element discretizations can be factored by triangular hierarchical matrices, SIAM J. Numer. Anal., 45 (2007), pp. 1472-1494.
    • (2007) SIAM J. Numer. Anal. , vol.45 , pp. 1472-1494
    • Bebendorf, M.1
  • 2
    • 21644458055 scopus 로고    scopus 로고
    • Efficient inversion of the Galerkin matrix of general second-order elliptic operators with nonsmooth coefficients
    • DOI 10.1090/S0025-5718-04-01716-8, PII S0025571804017168
    • M. Bebendorf, Efficient inversion of Galerkin matrices of general second-order elliptic differential operators with nonsmooth coefficients, Math. Comp., 74 (2005), pp. 1179-1199. (Pubitemid 40937959)
    • (2005) Mathematics of Computation , vol.74 , Issue.251 , pp. 1179-1199
    • Bebendorf, M.1
  • 3
  • 5
    • 61849133507 scopus 로고    scopus 로고
    • 2-matrices by hierarchical compression
    • 2-matrices by hierarchical compression, SIAM J. Sci. Comput., 31 (2009), pp. 1820-1839.
    • (2009) SIAM J. Sci. Comput. , vol.31 , pp. 1820-1839
    • Börm, S.1
  • 6
    • 24044501239 scopus 로고    scopus 로고
    • Hybrid cross approximation of integral operators
    • DOI 10.1007/s00211-005-0618-1
    • S. Börm and L. Grasedyck, Hybrid cross approximation of integral operators, Numer. Math., 101 (2005), pp. 221-249. (Pubitemid 41213100)
    • (2005) Numerische Mathematik , vol.101 , Issue.2 , pp. 221-249
    • Borm, S.1    Grasedyck, L.2
  • 7
    • 0037738931 scopus 로고    scopus 로고
    • Introduction to hierarchical matrices with applications
    • S. Börm, L. Grasedyck, and W. Hackbusch, Introduction to hierarchical matrices with applications, Eng. Anal. Bound. Elem, 27 (2003), pp. 405-422.
    • (2003) Eng. Anal. Bound. Elem , vol.27 , pp. 405-422
    • Börm, S.1    Grasedyck, L.2    Hackbusch, W.3
  • 8
    • 70350706800 scopus 로고    scopus 로고
    • 2-matrix-based integral-equation solver of reduced complexity and controlled accuracy for solving electrodynamic problems
    • 2-matrix-based integral-equation solver of reduced complexity and controlled accuracy for solving electrodynamic problems, IEEE Trans. Antennas and Propagation, 57 (2009), pp. 3147-3159.
    • (2009) IEEE Trans. Antennas and Propagation , vol.57 , pp. 3147-3159
    • Chai, W.1    Jiao, D.2
  • 11
    • 77956413970 scopus 로고    scopus 로고
    • On the numerical rank of the off-diagonal blocks of Schur complements of discretized elliptic PDEs
    • S. Chandrasekaran, P. Dewilde, M. Gu, and N. Somasunderam, On the numerical rank of the off-diagonal blocks of Schur complements of discretized elliptic PDEs, SIAM J.Matrix Anal. Appl., 31 (2010), pp. 2261-2290.
    • (2010) SIAM J.Matrix Anal. Appl. , vol.31 , pp. 2261-2290
    • Chandrasekaran, S.1    Dewilde, P.2    Gu, M.3    Somasunderam, N.4
  • 13
    • 33845283552 scopus 로고    scopus 로고
    • A fast adaptive solver for hierarchically semiseparable representations
    • DOI 10.1007/s10092-005-0103-3
    • S. Chandrasekaran, M. Gu, and W. Lyons, A fast adaptive solver for hierarchically semiseparable representations, Calcolo, 42 (2005), pp. 171-185. (Pubitemid 44870090)
    • (2005) Calcolo , vol.42 , Issue.3-4 , pp. 171-185
    • Chandrasekaran, S.1    Gu, M.2    Lyons, W.3
  • 14
    • 34547225006 scopus 로고    scopus 로고
    • A fast ULV decomposition solver for hierarchically semiseparable representations
    • DOI 10.1137/S0895479803436652
    • S. Chandrasekaran, M. Gu, and T. Pals, A fast ULV decomposition solver for hierarchically semiseparable representations, SIAM J. Matrix Anal. Appl., 28 (2006), pp. 603-622. (Pubitemid 47109352)
    • (2006) SIAM Journal on Matrix Analysis and Applications , vol.28 , Issue.3 , pp. 603-622
    • Chandrasekaran, S.1    Gu, M.2    Pals, T.3
  • 15
  • 16
    • 52949139479 scopus 로고    scopus 로고
    • A QR-based solver for rank structured matrices
    • S. Delvaux and M. Van Barel, A QR-based solver for rank structured matrices, SIAM. J. Matrix Anal. Appl., 30 (2008), pp. 464-490.
    • (2008) SIAM. J. Matrix Anal. Appl. , vol.30 , pp. 464-490
    • Delvaux, S.1    Van Barel, M.2
  • 17
    • 0003252789 scopus 로고    scopus 로고
    • Applied numerical linear algebra
    • Philadelphia
    • J. W. Demmel, Applied Numerical Linear Algebra, SIAM, Philadelphia, 1997.
    • (1997) SIAM
    • Demmel, J.W.1
  • 18
    • 0020822138 scopus 로고
    • The multifrontal solution of indefinite sparse symmetric linear equations
    • DOI 10.1145/356044.356047
    • I. S. Duff and J. K. Reid, The multifrontal solution of indefinite sparse symmetric linear equations, ACM Trans. Math. Software, 9 (1983), pp. 302-325. (Pubitemid 14454003)
    • (1983) ACM Transactions on Mathematical Software , vol.9 , Issue.3 , pp. 302-325
    • Duff, I.S.1    Reid, J.K.2
  • 19
    • 20344378938 scopus 로고    scopus 로고
    • The QR iteration method for Hermitian quasiseparable matrices of an arbitrary order
    • DOI 10.1016/j.laa.2005.02.037, PII S0024379505001369
    • Y. Eidelman, I. Gohberg, and V. Olshevsky, The QR iteration method for Hermitian quasiseparable matrices of an arbitrary order, Linear Algebra Appl., 404 (2005), pp. 305-324. (Pubitemid 40786536)
    • (2005) Linear Algebra and Its Applications , vol.404 , Issue.1-3 , pp. 305-324
    • Eidelman, Y.1    Gohberg, I.2    Olshevsky, V.3
  • 20
    • 79551482511 scopus 로고    scopus 로고
    • Sweeping preconditioner for the Helmholtz equation: Hierarchical matrix representation
    • B. Engquist and L. Ying, Sweeping preconditioner for the Helmholtz equation: Hierarchical matrix representation, Comm. Pure Appl. Math., 64 (2011), pp. 697-735.
    • (2011) Comm. Pure Appl. Math. , vol.64 , pp. 697-735
    • Engquist, B.1    Ying, L.2
  • 21
    • 0001883973 scopus 로고
    • Hankel and Loewner matrices
    • M. Fiedler, Hankel and Loewner matrices, Linear Algebra Appl., 58 (1984), pp. 75-95.
    • (1984) Linear Algebra Appl. , vol.58 , pp. 75-95
    • Fiedler, M.1
  • 23
    • 0004236492 scopus 로고    scopus 로고
    • 3rd ed., The Johns Hopkins University Press, Baltimore, MD
    • G. H. Golub and C. V. Loan, Matrix Computations, 3rd ed., The Johns Hopkins University Press, Baltimore, MD, 1996.
    • (1996) Matrix Computations
    • Golub, G.H.1    Loan, C.V.2
  • 24
    • 47849120618 scopus 로고    scopus 로고
    • Parallel black box H-LU preconditioning for elliptic boundary value problems
    • L. Grasedyck, R. Kriemann, and S. Le Borne, Parallel black box H-LU preconditioning for elliptic boundary value problems, Comput. Vis. Sci., 11 (2008), pp. 273-291.
    • (2008) Comput. Vis. Sci. , vol.11 , pp. 273-291
    • Grasedyck, L.1    Kriemann, R.2    Le Borne, S.3
  • 25
    • 84867289869 scopus 로고    scopus 로고
    • A numerically stable and superfast algorithm for solving Toeplitz systems of linear equations
    • Monterey, CA
    • M. Gu and J. Xia, A numerically stable and superfast algorithm for solving Toeplitz systems of linear equations, in Proceedings of the SIAM Conference on Applied Linear Algebra (LA09), Monterey, CA, 2009.
    • (2009) Proceedings of the SIAM Conference on Applied Linear Algebra (LA09)
    • Gu, M.1    Xia, J.2
  • 26
    • 0032637646 scopus 로고    scopus 로고
    • A sparse matrix arithmetic based on H-matrices. Part I: Introduction to H-matrices
    • W. Hackbusch, A sparse matrix arithmetic based on H-matrices. Part I: Introduction to H-matrices, Computing, 62 (1999), pp. 89-108.
    • (1999) Computing , vol.62 , pp. 89-108
    • Hackbusch, W.1
  • 29
    • 0002133258 scopus 로고    scopus 로고
    • A sparse H-matrix arithmetic. Part II: Application to multi-dimensional problems
    • W. Hackbusch and B. N. Khoromskij, A sparse H-matrix arithmetic. Part II: Application to multi-dimensional problems, Computing, 64 (2000), pp. 21-47.
    • (2000) Computing , vol.64 , pp. 21-47
    • Hackbusch, W.1    Khoromskij, B.N.2
  • 31
    • 79952901900 scopus 로고    scopus 로고
    • Fast construction of hierarchical matrix representation from matrix-vector multiplication
    • L. Lin, J. Lu, and L. Ying, Fast construction of hierarchical matrix representation from matrix-vector multiplication, J. Comput. Phys., 230 (2011), pp. 4071-4087.
    • (2011) J. Comput. Phys. , vol.230 , pp. 4071-4087
    • Lin, L.1    Lu, J.2    Ying, L.3
  • 32
    • 0026840122 scopus 로고
    • The multifrontal method for sparse matrix solution: Theory and practice
    • J. W. H. Liu, The multifrontal method for sparse matrix solution: Theory and practice, SIAM Rev., 34 (1992), pp. 82-109.
    • (1992) SIAM Rev. , vol.34 , pp. 82-109
    • Liu, J.W.H.1
  • 34
    • 84856259101 scopus 로고    scopus 로고
    • A fast randomized algorithm for computing a hierarchically semiseparable representation of a matrix
    • P. G. Martinsson, A fast randomized algorithm for computing a hierarchically semiseparable representation of a matrix, SIAM J. Matrix Anal. Appl., 32 (2011), pp. 1251-1274.
    • (2011) SIAM J. Matrix Anal. Appl. , vol.32 , pp. 1251-1274
    • Martinsson, P.G.1
  • 35
    • 64949123438 scopus 로고    scopus 로고
    • A fast direct solver for a class of elliptic partial differential equations
    • P. G. Martinsson, A fast direct solver for a class of elliptic partial differential equations, J. Sci. Comput., 38 (2009), pp. 316-330.
    • (2009) J. Sci. Comput. , vol.38 , pp. 316-330
    • Martinsson, P.G.1
  • 36
    • 16844371628 scopus 로고    scopus 로고
    • A fast direct solver for boundary integral equations in two dimensions
    • P. G. Martinsson and V. Rokhlin, A fast direct solver for boundary integral equations in two dimensions, J. Comput. Phys., 205 (2005), pp. 1-23.
    • (2005) J. Comput. Phys. , vol.205 , pp. 1-23
    • Martinsson, P.G.1    Rokhlin, V.2
  • 37
    • 26444607857 scopus 로고    scopus 로고
    • A fast algorithm for the inversion of general Toeplitz matrices
    • DOI 10.1016/j.camwa.2005.03.011, PII S0898122105003147
    • P. G. Martinsson, V. Rokhlin, and M. Tygert, A fast algorithm for the inversion of general Toeplitz matrices, Comput. Math. Appl., 50 (2005), pp. 741-752. (Pubitemid 41427157)
    • (2005) Computers and Mathematics with Applications , vol.50 , Issue.5-6 , pp. 741-752
    • Martinsson, P.G.1    Rokhlin, V.2    Tygert, M.3
  • 38
    • 84966229708 scopus 로고
    • On computations with dense structured matrices
    • V. Pan, On computations with dense structured matrices, Math. Comp., 55 (1990), pp. 179-190.
    • (1990) Math. Comp. , vol.55 , pp. 179-190
    • Pan, V.1
  • 39
    • 79251529275 scopus 로고    scopus 로고
    • Algorithms to solve hierarchically semiseparable systems
    • Birkhäuser, Basel
    • Z. Sheng, P. Dewilde, and S. Chandrasekaran, Algorithms to Solve Hierarchically Semiseparable Systems, Oper. Theory Adv. Appl. 176, Birkhäuser, Basel, 2007, pp. 255-294.
    • (2007) Oper. Theory Adv. Appl. 176 , pp. 255-294
    • Sheng, Z.1    Dewilde, P.2    Chandrasekaran, S.3
  • 40
    • 33845343264 scopus 로고    scopus 로고
    • A bibliography on semiseparable matrices
    • DOI 10.1007/s10092-005-0107-z
    • R. Vandebril, M. Van Barel, G. Golub, and N. Mastronardi, A bibliography on semiseparable matrices, Calcolo, 42 (2005), pp. 249-270. (Pubitemid 44870094)
    • (2005) Calcolo , vol.42 , Issue.3-4 , pp. 249-270
    • Vandebril, R.1    Van Barel, M.2    Golub, G.3    Mastronardi, N.4
  • 41
    • 77956417875 scopus 로고    scopus 로고
    • Superfast multifrontal method for large structured linear systems of equations
    • J. Xia, S. Chandrasekaran, M. Gu, and X. S. Li, Superfast multifrontal method for large structured linear systems of equations, SIAM J. Matrix Anal. Appl., 31 (2009), pp. 1382-1411.
    • (2009) SIAM J. Matrix Anal. Appl. , vol.31 , pp. 1382-1411
    • Xia, J.1    Chandrasekaran, S.2    Gu, M.3    Li, X.S.4
  • 42
  • 43
    • 79251500987 scopus 로고    scopus 로고
    • Robust approximate Cholesky factorization of rank-structured symmetric positive definite matrices
    • J. Xia and M. Gu, Robust approximate Cholesky factorization of rank-structured symmetric positive definite matrices, SIAM J. Matrix Anal. Appl., 31 (2010), pp. 2899-2920.
    • (2010) SIAM J. Matrix Anal. Appl. , vol.31 , pp. 2899-2920
    • Xia, J.1    Gu, M.2


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.