메뉴 건너뛰기




Volumn 6, Issue 2, 2006, Pages 194-220

Structured Rank-(r1, …, rd) Decomposition of Function-related Tensors in ℝ

Author keywords

H matrices; higher order tensors; Kronecker products; matrix valued functions; multidimensional integral operators; Newton potential; structured matrices

Indexed keywords


EID: 34447532966     PISSN: 16094840     EISSN: 16099389     Source Type: Journal    
DOI: 10.2478/cmam-2006-0010     Document Type: Article
Times cited : (70)

References (42)
  • 1
    • 0036679114 scopus 로고    scopus 로고
    • Numerical operator calculus in higher dimensions
    • G. Beylkin, M. M. Mohlenkamp, Numerical operator calculus in higher dimensions, PNAS, 99(2002), no. 16, pp. 10246-10251.
    • (2002) PNAS, 99 , Issue.16 , pp. 10246-10251
    • Beylkin, G.1    Mohlenkamp, M.M.2
  • 2
    • 26444558083 scopus 로고    scopus 로고
    • Approximation of 1/x by exponential sums in [1, to)
    • G. Braess and W. Hackbusch, Approximation of 1/x by exponential sums in [1, to), IMA J. Numer. Anal. 25 (2005), no. 4, pp. 685-697.
    • (2005) IMA J. Numer. Anal. , vol.25 , Issue.4 , pp. 685-697
    • Braess, G.1    Hackbusch, W.2
  • 3
    • 34250499792 scopus 로고
    • Analysis of individual differences in multidimensional scaling via an N-way generalization of “Eckart-Young” decomposition
    • J. D. Carrol and J. Chang, Analysis of individual differences in multidimensional scaling via an N-way generalization of “Eckart-Young” decomposition, Psychometrika, 35 (1970), pp. 283-319.
    • (1970) Psychometrika , vol.35 , pp. 283-319
    • Carrol, J.D.1    Chang, J.2
  • 5
    • 0034144761 scopus 로고    scopus 로고
    • On the best rank-1 and rank-(R1, …, RN approximation of higher-order tensors
    • L. De Lathauwer, B. De Moor, J. Vandewalle, On the best rank-1 and rank-(R1, …, RN) approximation of higher-order tensors, SIAM J. Matrix Anal. Appl., 21 (2000), pp. 1324-1342.
    • (2000) SIAM J. Matrix Anal. Appl , vol.21 , pp. 1324-1342
    • De Lathauwer, L.1    De Moor, B.2    Vandewalle, J.3
  • 6
    • 17444379662 scopus 로고    scopus 로고
    • Computation of the canoninical decomposition by means of a simultaneous generalized Schur decomposition
    • L. De Lathauwer, B. DeMoor, J. Vandewalle, Computation of the canoninical decomposition by means of a simultaneous generalized Schur decomposition, SIAM J. Matrix Anal. Appl., 26 (2004), pp. 295-327.
    • (2004) SIAM J. Matrix Anal. Appl , vol.26 , pp. 295-327
    • De Lathauwer, L.1    DeMoor, B.2    Vandewalle, J.3
  • 7
    • 34447551264 scopus 로고    scopus 로고
    • Concept of data-sparse tensor-product approximation in many-particle models
    • (in preparation).
    • H.-J. Flad, W. Hackbusch, B. N. Khoromskij, and R. Schneider, Concept of data-sparse tensor-product approximation in many-particle models (in preparation).
    • Flad, H.-J.1    Hackbusch, W.2    Khoromskij, B.N.3    Schneider, R.4
  • 8
    • 0037042658 scopus 로고    scopus 로고
    • Wavelet approximation of correlated wave-functions. I. Basics
    • H.-J. Flad, W. Hackbusch, D. Kolb, and R. Schneider, Wavelet approximation of correlated wave-functions. I. Basics, J. Chem. Phys., 116 (2002), pp. 9641-9657.
    • (2002) J. Chem. Phys , vol.116 , pp. 9641-9657
    • Flad, H.-J.1    Hackbusch, W.2    Kolb, D.3    Schneider, R.4
  • 9
    • 2942644816 scopus 로고    scopus 로고
    • Data-sparse approximation to operator-valued functions of elliptic operators
    • I. P. Gavrilyuk, W. Hackbusch, and B. N. Khoromskij, Data-sparse approximation to operator-valued functions of elliptic operators. Math. Comp., 73 (2003), pp. 1297-1324.
    • (2003) Math. Comp , vol.73 , pp. 1297-1324
    • Gavrilyuk, I.P.1    Hackbusch, W.2    Khoromskij, B.N.3
  • 10
    • 14944359407 scopus 로고    scopus 로고
    • Data-sparse approximation to a class of operator-valued functions
    • I. P. Gavrilyuk, W. Hackbusch, and B. N. Khoromskij, Data-sparse approximation to a class of operator-valued functions. Math. Comp., 74 (2005), pp. 681-708.
    • (2005) Math. Comp , vol.74 , pp. 681-708
    • Gavrilyuk, I.P.1    Hackbusch, W.2    Khoromskij, B.N.3
  • 11
    • 17144410769 scopus 로고    scopus 로고
    • Tensor-product approximation to elliptic and parabolic solution operators in higher dimensions
    • I. P. Gavrilyuk, W. Hackbusch, and B. N. Khoromskij, Tensor-product approximation to elliptic and parabolic solution operators in higher dimensions, Computing, 74 (2005), pp. 131-157.
    • (2005) Computing , vol.74 , pp. 131-157
    • Gavrilyuk, I.P.1    Hackbusch, W.2    Khoromskij, B.N.3
  • 12
    • 33749034221 scopus 로고    scopus 로고
    • Exponentially convergent algorithms for the operator exponential with applications to inhomegeneous problems in Banach spaces
    • I. P. Gavrilyuk and V. L. Makarov, Exponentially convergent algorithms for the operator exponential with applications to inhomegeneous problems in Banach spaces, SIAM J. on Numer. Anal., 43 (2005), no. 5, pp. 2144-2171.
    • (2005) SIAM J. on Numer. Anal. , vol.43 , Issue.5 , pp. 2144-2171
    • Gavrilyuk, I.P.1    Makarov, V.L.2
  • 13
    • 14944374194 scopus 로고    scopus 로고
    • Algorithms without accuracy saturation for evolution equations in Hilbert and Banach spaces
    • I. P. Gavrilyuk and V. L. Makarov, Algorithms without accuracy saturation for evolution equations in Hilbert and Banach spaces, Math. Comp., 74 (2005), pp. 555-583.
    • (2005) Math. Comp , vol.74 , pp. 555-583
    • Gavrilyuk, I.P.1    Makarov, V.L.2
  • 14
    • 0004236492 scopus 로고    scopus 로고
    • Matrix Computations
    • Johns Hopkins University Press, Baltimore, MD
    • G. H. Golub, C. F. Van Loan, Matrix Computations, Johns Hopkins University Press, Baltimore, MD, 1996.
    • (1996)
    • Golub, G.H.1    Van Loan, C.F.2
  • 15
    • 0141829814 scopus 로고    scopus 로고
    • Construction and arithmetics of H-matrices
    • L. Grasedyck and W. Hackbusch, Construction and arithmetics of H-matrices, Computing, 70 (2003), pp. 295-334.
    • (2003) Computing , vol.70 , pp. 295-334
    • Grasedyck, L.1    Hackbusch, W.2
  • 16
    • 0034337983 scopus 로고    scopus 로고
    • Optimized tensor-product approximation spaces
    • M. Griebel and S. Knapek, Optimized tensor-product approximation spaces, Constr. Approx., 16 (2000), pp. 303-332.
    • (2000) Constr. Approx , vol.16 , pp. 303-332
    • Griebel, M.1    Knapek, S.2
  • 17
    • 0032637646 scopus 로고    scopus 로고
    • A sparse matrix arithmetic based on H-matrices. I. Introduction to H-matrices
    • W. Hackbusch, A sparse matrix arithmetic based on H-matrices. I. Introduction to H-matrices, Computing, 62 (1999), pp. 89-108.
    • (1999) Computing , vol.62 , pp. 89-108
    • Hackbusch, W.1
  • 18
    • 0002133258 scopus 로고    scopus 로고
    • A sparse H-matrix arithmetic. II. Application to multidimensional problems
    • W. Hackbusch and B. N. Khoromskij, A sparse H-matrix arithmetic. II. Application to multidimensional problems, Computing, 64 (2000), pp. 21-47.
    • (2000) Computing , vol.64 , pp. 21-47
    • Hackbusch, W.1    Khoromskij, B.N.2
  • 19
    • 0034509772 scopus 로고    scopus 로고
    • A sparse H-matrix arithmetic: General complexity estimates
    • W. Hackbusch and B. N. Khoromskij, A sparse H-matrix arithmetic: General complexity estimates, J. Comp. Appl. Math., 125 (2000), pp. 479-501.
    • (2000) J. Comp. Appl. Math. , vol.125 , pp. 479-501
    • Hackbusch, W.1    Khoromskij, B.N.2
  • 20
    • 29144513721 scopus 로고    scopus 로고
    • Low-rank Kronecker product approximation to multidimensional nonlocal operators. Part I. Separable approximation of multivariate functions
    • W. Hackbusch and B. N. Khoromskij, Low-rank Kronecker product approximation to multidimensional nonlocal operators. Part I. Separable approximation of multivariate functions, Computing, 76 (2006), pp. 177-202.
    • (2006) Computing , vol.76 , pp. 177-202
    • Hackbusch, W.1    Khoromskij, B.N.2
  • 21
    • 29144481646 scopus 로고    scopus 로고
    • Low-rank Kronecker product approximation to multidimensional nonlocal operators. Part II. HKT representations of certain operators
    • W. Hackbusch and B. N. Khoromskij, Low-rank Kronecker product approximation to multidimensional nonlocal operators. Part II. HKT representations of certain operators, Computing, 76 (2006), pp. 203-225.
    • (2006) Computing , vol.76 , pp. 203-225
    • Hackbusch, W.1    Khoromskij, B.N.2
  • 22
    • 21244436795 scopus 로고    scopus 로고
    • Hierarchical Kronecker tensor-product approximations
    • W. Hackbusch, B. N. Khoromskij and E. E. Tyrtyshnikov, Hierarchical Kronecker tensor-product approximations, J. Numer. Math., 13 (2005), pp. 119-156.
    • (2005) J. Numer. Math , vol.13 , pp. 119-156
    • Hackbusch, W.1    Khoromskij, B.N.2    Tyrtyshnikov, E.E.3
  • 24
    • 0002740437 scopus 로고
    • Foundation of the PARAFAC procedure: Model and conditions for an “explanatory” multimode factor analysis
    • R. Harshman, Foundation of the PARAFAC procedure: Model and conditions for an “explanatory” multimode factor analysis, UCLA Working Papers in Phonetics, 16 (1970), pp. 1-84.
    • (1970) UCLA Working Papers in Phonetics , vol.16 , pp. 1-84
    • Harshman, R.1
  • 25
  • 26
    • 84900554198 scopus 로고    scopus 로고
    • An Introduction to Structured Tensor-Product Representation of Discrete Nonlocal Operators
    • Lecture Notes 27, MPI MIS, Leipzig
    • B. N. Khoromskij, An Introduction to Structured Tensor-Product Representation of Discrete Nonlocal Operators, Lecture Notes 27, MPI MIS, Leipzig, 2005.
    • (2005)
    • Khoromskij, B.N.1
  • 27
    • 85025270057 scopus 로고    scopus 로고
    • Dynamical Low Rank Approximation
    • Preprint Uni. Tuebingen
    • O. Koch and Ch. Lubich, Dynamical Low Rank Approximation, Preprint Uni. Tuebingen, 2005.
    • (2005)
    • Koch, O.1    Lubich, C.2
  • 28
    • 0036055687 scopus 로고    scopus 로고
    • Orthogonal tensor decompositions
    • T. Kolda, Orthogonal tensor decompositions, SIAM J. Matrix Anal. Appl., 23 (2001), pp. 243-255.
    • (2001) SIAM J. Matrix Anal. Appl , vol.23 , pp. 243-255
    • Kolda, T.1
  • 29
    • 48749101457 scopus 로고
    • Three-way arrays: rank and uniqueness of trilinear decompositions, with applications to arithmetic complexity and statistics
    • J. B. Kruskal, Three-way arrays: rank and uniqueness of trilinear decompositions, with applications to arithmetic complexity and statistics, Linear Algebra Appl., 18 (1977), pp. 95-138.
    • (1977) Linear Algebra Appl , vol.18 , pp. 95-138
    • Kruskal, J.B.1
  • 30
    • 0001837391 scopus 로고
    • Rank, decomposition, and uniqueness for 3-way and N-way arrays
    • R. Coppi and S. Bolasco, eds., North-Holland, Amsterdam
    • J. B. Kruskal, Rank, decomposition, and uniqueness for 3-way and N-way arrays, In Multiway Data Analysis, R. Coppi and S. Bolasco, eds., North-Holland, Amsterdam, 1989, pp. 7-18.
    • (1989) Multiway Data Analysis , pp. 7-18
    • Kruskal, J.B.1
  • 31
    • 85025296879 scopus 로고    scopus 로고
    • emphA spectral order method for inverting sectorial Laplace transform, SINUM, to appear.
    • M. Lopez-Fernandez, C. Palencia and A. Schadle, emphA spectral order method for inverting sectorial Laplace transform, SINUM 2006, to appear.
    • (2006)
    • Lopez-Fernandez, M.1    Palencia, C.2    Schadle, A.3
  • 32
    • 14944376375 scopus 로고    scopus 로고
    • On Variational Approximations in Quantum Molecular Dynamics
    • Ch. Lubich, On Variational Approximations in Quantum Molecular Dynamics, Math. Comp., 74 (2005), pp. 765-779.
    • (2005) Math. Comp , vol.74 , pp. 765-779
    • Lubich, C.1
  • 33
    • 1342326246 scopus 로고    scopus 로고
    • Numerical solution of parabolic equations in high dimensions
    • T. von Petersdorf and Ch. Schwab, Numerical solution of parabolic equations in high dimensions, M2AN, 38 (2004), no. 1, pp. 93-127.
    • (2004) M2AN , vol.38 , Issue.1 , pp. 93-127
    • von Petersdorf, T.1    Schwab, C.2
  • 36
    • 85025277491 scopus 로고
    • Numerical methods based on Sinc and analytic functions
    • F. Stenger, Numerical methods based on Sinc and analytic functions, Springer - Verlag, Heidelberg, 1993.
    • (1993) Springer - Verlag, Heidelberg
    • Stenger, F.1
  • 38
    • 0013953617 scopus 로고
    • Some mathematical notes on three-mode factor analysis
    • L. R. Tucker, Some mathematical notes on three-mode factor analysis, Psychometrika, 31 (1966), pp. 279-311.
    • (1966) Psychometrika , vol.31 , pp. 279-311
    • Tucker, L.R.1
  • 39
    • 0142055272 scopus 로고    scopus 로고
    • Tensor approximations of matrices generated by asymptotically smooth functions
    • Sbornik: Mathematics
    • E. E. Tyrtyshnikov, Tensor approximations of matrices generated by asymptotically smooth functions, Sbornik: Mathematics 194 (2003), no. 5-6, pp. 941-954
    • (2003) , vol.194 , Issue.5-6 , pp. 941-954
    • Tyrtyshnikov, E.E.1
  • 40
    • 0742320626 scopus 로고    scopus 로고
    • Kronecker-product approximations for some function-related matrices
    • E. E. Tyrtyshnikov, Kronecker-product approximations for some function-related matrices, Linear Algebra Appl., 379 (2004), pp. 423-437.
    • (2004) Linear Algebra Appl , vol.379 , pp. 423-437
    • Tyrtyshnikov, E.E.1
  • 41
    • 2942635358 scopus 로고    scopus 로고
    • On the regularity of the electronic Schrödinger equation in Hilbert spaces of mixed derivatives
    • H. Yserentant, On the regularity of the electronic Schrödinger equation in Hilbert spaces of mixed derivatives, Numer. Math., 98 (2004), pp. 731-759.
    • (2004) Numer. Math , vol.98 , pp. 731-759
    • Yserentant, H.1
  • 42
    • 0036056838 scopus 로고    scopus 로고
    • Rank-one approximation to high order tensors
    • T. Zang and G. Golub, Rank-one approximation to high order tensors, SIAM J. Matrix Anal. Appl., 23 (2001), pp. 534-550.
    • (2001) SIAM J. Matrix Anal. Appl. , vol.23 , pp. 534-550
    • Zang, T.1    Golub, G.2


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.