메뉴 건너뛰기




Volumn 185, Issue 4, 2014, Pages 1207-1216

Computation of extreme eigenvalues in higher dimensions using block tensor train format

Author keywords

DMRG; High dimensional problems; Low lying eigenstates; MPS; Tensor train format

Indexed keywords


EID: 84894609844     PISSN: 00104655     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cpc.2013.12.017     Document Type: Article
Times cited : (107)

References (52)
  • 1
    • 68649096448 scopus 로고    scopus 로고
    • Tensor decompositions and applications
    • T.G. Kolda, and B.W. Bader Tensor decompositions and applications SIAM Rev. 51 2009 455 500
    • (2009) SIAM Rev. , vol.51 , pp. 455-500
    • Kolda, T.G.1    Bader, B.W.2
  • 2
    • 33751260933 scopus 로고    scopus 로고
    • A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids
    • A. Ammar, B. Mokdad, F. Chinesta, and R. Keunings A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids J. Non-Newton. Fluid Mech. 139 2006 153 176
    • (2006) J. Non-Newton. Fluid Mech. , vol.139 , pp. 153-176
    • Ammar, A.1    Mokdad, B.2    Chinesta, F.3    Keunings, R.4
  • 3
    • 70449574498 scopus 로고    scopus 로고
    • Results and questions on a nonlinear approximation approach for solving high-dimensional partial differential equations
    • C. Le Bris, T. Leliévre, and Y. Maday Results and questions on a nonlinear approximation approach for solving high-dimensional partial differential equations Constr. Approx. 30 2009 621 651
    • (2009) Constr. Approx. , vol.30 , pp. 621-651
    • Le Bris, C.1    Leliévre, T.2    Maday, Y.3
  • 5
    • 84897586218 scopus 로고    scopus 로고
    • Multigrid accelerated tensor approximation of function related multidimensional arrays
    • B.N. Khoromskij, and V. Khoromskaia Multigrid accelerated tensor approximation of function related multidimensional arrays SIAM J. Sci. Comput. 31 2009 3002 3026
    • (2009) SIAM J. Sci. Comput. , vol.31 , pp. 3002-3026
    • Khoromskij, B.N.1    Khoromskaia, V.2
  • 8
    • 3442895828 scopus 로고
    • Density matrix formulation for quantum renormalization groups
    • S.R. White Density matrix formulation for quantum renormalization groups Phys. Rev. Lett. 69 1992 2863 2866
    • (1992) Phys. Rev. Lett. , vol.69 , pp. 2863-2866
    • White, S.R.1
  • 9
    • 20044389808 scopus 로고
    • Density-matrix algorithms for quantum renormalization groups
    • S.R. White Density-matrix algorithms for quantum renormalization groups Phys. Rev. B 48 1993 10345 10356
    • (1993) Phys. Rev. B , vol.48 , pp. 10345-10356
    • White, S.R.1
  • 10
  • 11
    • 84956137131 scopus 로고
    • Matrix product ground states for one-dimensional spin-1 quantum antiferromagnets
    • A. Klümper, A. Schadschneider, and J. Zittartz Matrix product ground states for one-dimensional spin-1 quantum antiferromagnets Europhys. Lett. 24 1993 293 297
    • (1993) Europhys. Lett. , vol.24 , pp. 293-297
    • Klümper, A.1    Schadschneider, A.2    Zittartz, J.3
  • 12
    • 78650755970 scopus 로고    scopus 로고
    • The density-matrix renormalization group in the age of matrix product states
    • U. Schollwöck The density-matrix renormalization group in the age of matrix product states Ann. Physics 326 2011 96 192
    • (2011) Ann. Physics , vol.326 , pp. 96-192
    • Schollwöck, U.1
  • 13
    • 70449528138 scopus 로고    scopus 로고
    • Breaking the curse of dimensionality, or how to use SVD in many dimensions
    • I.V. Oseledets, and E.E. Tyrtyshnikov Breaking the curse of dimensionality, or how to use SVD in many dimensions SIAM J. Sci. Comput. 31 2009 3744 3759
    • (2009) SIAM J. Sci. Comput. , vol.31 , pp. 3744-3759
    • Oseledets, I.V.1    Tyrtyshnikov, E.E.2
  • 14
    • 80053896203 scopus 로고    scopus 로고
    • Tensor-train decomposition
    • I.V. Oseledets Tensor-train decomposition SIAM J. Sci. Comput. 33 2011 2295 2317
    • (2011) SIAM J. Sci. Comput. , vol.33 , pp. 2295-2317
    • Oseledets, I.V.1
  • 15
    • 77952289640 scopus 로고    scopus 로고
    • A new scheme for the tensor representation
    • W. Hackbusch, and S. Kühn A new scheme for the tensor representation J. Fourier Anal. Appl. 15 2009 706 722
    • (2009) J. Fourier Anal. Appl. , vol.15 , pp. 706-722
    • Hackbusch, W.1    Kühn, S.2
  • 16
    • 77956032667 scopus 로고    scopus 로고
    • Hierarchical singular value decomposition of tensors
    • L. Grasedyck Hierarchical singular value decomposition of tensors SIAM J. Matrix Anal. Appl. 31 2010 2029 2054
    • (2010) SIAM J. Matrix Anal. Appl. , vol.31 , pp. 2029-2054
    • Grasedyck, L.1
  • 17
    • 84861379430 scopus 로고    scopus 로고
    • The alternating linear scheme for tensor optimization in the tensor train format
    • S. Holtz, T. Rohwedder, and R. Schneider The alternating linear scheme for tensor optimization in the tensor train format SIAM J. Sci. Comput. 34 2012 A683 A713
    • (2012) SIAM J. Sci. Comput. , vol.34
    • Holtz, S.1    Rohwedder, T.2    Schneider, R.3
  • 18
    • 84869825516 scopus 로고    scopus 로고
    • Solution of linear systems and matrix inversion in the TT-format
    • S.V. Dolgov, and I.V. Oseledets Solution of linear systems and matrix inversion in the TT-format SIAM J. Sci. Comput. 34 2012 A2718 A2739
    • (2012) SIAM J. Sci. Comput. , vol.34
    • Dolgov, S.V.1    Oseledets, I.V.2
  • 20
    • 84871569236 scopus 로고    scopus 로고
    • Fast solution of multi-dimensional parabolic problems in the tensor train/quantized tensor train-format with initial application to the Fokker-Planck equation
    • S.V. Dolgov, B.N. Khoromskij, and I.V. Oseledets Fast solution of multi-dimensional parabolic problems in the tensor train/quantized tensor train-format with initial application to the Fokker-Planck equation SIAM J. Sci. Comput. 34 2012 A3016 A3038
    • (2012) SIAM J. Sci. Comput. , vol.34
    • Dolgov, S.V.1    Khoromskij, B.N.2    Oseledets, I.V.3
  • 22
    • 81355160934 scopus 로고    scopus 로고
    • Tensor conjugate-gradient-type method for Rayleigh quotient minimization in block QTT-format
    • O.S. Lebedeva Tensor conjugate-gradient-type method for Rayleigh quotient minimization in block QTT-format Russ. J. Numer. Anal. Math. Modelling 26 2011 465 489
    • (2011) Russ. J. Numer. Anal. Math. Modelling , vol.26 , pp. 465-489
    • Lebedeva, O.S.1
  • 25
    • 77955278344 scopus 로고    scopus 로고
    • Fast and accurate tensor approximation of multivariate convolution with linear scaling in dimension
    • B.N. Khoromskij Fast and accurate tensor approximation of multivariate convolution with linear scaling in dimension J. Comput. Appl. Math. 234 2010 3122 3139
    • (2010) J. Comput. Appl. Math. , vol.234 , pp. 3122-3139
    • Khoromskij, B.N.1
  • 26
    • 80053999286 scopus 로고    scopus 로고
    • Tensorisation of vectors and their efficient convolution
    • W. Hackbusch Tensorisation of vectors and their efficient convolution Numer. Math. 119 2011 465 488
    • (2011) Numer. Math. , vol.119 , pp. 465-488
    • Hackbusch, W.1
  • 27
    • 84884918548 scopus 로고    scopus 로고
    • Multilevel Toeplitz matrices generated by tensor-structured vectors and convolution with logarithmic complexity
    • V. Kazeev, B. Khoromskij, and E. Tyrtyshnikov Multilevel Toeplitz matrices generated by tensor-structured vectors and convolution with logarithmic complexity SIAM J. Sci. Comput. 35 2013 A1511 A1536
    • (2013) SIAM J. Sci. Comput. , vol.35
    • Kazeev, V.1    Khoromskij, B.2    Tyrtyshnikov, E.3
  • 30
    • 79960994573 scopus 로고    scopus 로고
    • O (d log n) -Quantics approximation of N-d tensors in high-dimensional numerical modeling
    • B.N. Khoromskij O (d log n) -Quantics approximation of N-d tensors in high-dimensional numerical modeling Constr. Approx. 34 2011 257 280
    • (2011) Constr. Approx. , vol.34 , pp. 257-280
    • Khoromskij, B.N.1
  • 32
    • 84880561200 scopus 로고    scopus 로고
    • Two-level QTT-Tucker format for optimized tensor calculus
    • S. Dolgov, and B. Khoromskij Two-level QTT-Tucker format for optimized tensor calculus SIAM J. Matrix Anal. Appl. 34 2013 593 623
    • (2013) SIAM J. Matrix Anal. Appl. , vol.34 , pp. 593-623
    • Dolgov, S.1    Khoromskij, B.2
  • 33
    • 34447513756 scopus 로고    scopus 로고
    • Low rank Tucker-type tensor approximation to classical potentials
    • B.N. Khoromskij, and V. Khoromskaia Low rank Tucker-type tensor approximation to classical potentials Cent. Eur. J. Math. 5 2007 523 550
    • (2007) Cent. Eur. J. Math. , vol.5 , pp. 523-550
    • Khoromskij, B.N.1    Khoromskaia, V.2
  • 36
    • 79952297523 scopus 로고    scopus 로고
    • Numerical solution of the Hartree-Fock equation in multilevel tensor-structured format
    • B.N. Khoromskij, V. Khoromskaia, and H.-J. Flad Numerical solution of the Hartree-Fock equation in multilevel tensor-structured format SIAM J. Sci. Comput. 33 2011 45 65
    • (2011) SIAM J. Sci. Comput. , vol.33 , pp. 45-65
    • Khoromskij, B.N.1    Khoromskaia, V.2    Flad, H.-J.3
  • 37
    • 84880611682 scopus 로고    scopus 로고
    • Tensor-structured factorized calculation of two-electron integrals in a general basis
    • V. Khoromskaia, B.N. Khoromskij, and R. Schneider Tensor-structured factorized calculation of two-electron integrals in a general basis SIAM J. Sci. Comput. 35 2013 A987 A1010
    • (2013) SIAM J. Sci. Comput. , vol.35
    • Khoromskaia, V.1    Khoromskij, B.N.2    Schneider, R.3
  • 38
    • 35949034896 scopus 로고
    • The renormalization group: Critical phenomena and the Kondo problem
    • K.G. Wilson The renormalization group: critical phenomena and the Kondo problem Rev. Modern Phys. 47 1975 773 840
    • (1975) Rev. Modern Phys. , vol.47 , pp. 773-840
    • Wilson, K.G.1
  • 39
    • 84856895584 scopus 로고    scopus 로고
    • Variational numerical renormalization group: Bridging the gap between NRG and density matrix renormalization group
    • I. Pižorn, and F. Verstraete Variational numerical renormalization group: bridging the gap between NRG and density matrix renormalization group Phys. Rev. Lett. 108 2012
    • (2012) Phys. Rev. Lett. , vol.108
    • Pižorn, I.1    Verstraete, F.2
  • 41
    • 84880562542 scopus 로고    scopus 로고
    • Local convergence of alternating schemes for optimization of convex problems in the TT format
    • T. Rohwedder, and A. Uschmajew Local convergence of alternating schemes for optimization of convex problems in the TT format SIAM J. Numer. Anal. 51 2013 1134 1162
    • (2013) SIAM J. Numer. Anal. , vol.51 , pp. 1134-1162
    • Rohwedder, T.1    Uschmajew, A.2
  • 42
    • 80053929473 scopus 로고    scopus 로고
    • DMRG approach to fast linear algebra in the TT-format
    • I.V. Oseledets DMRG approach to fast linear algebra in the TT-format Comput. Methods Appl. Math. 11 2011 382 393
    • (2011) Comput. Methods Appl. Math. , vol.11 , pp. 382-393
    • Oseledets, I.V.1
  • 43
    • 0036223489 scopus 로고    scopus 로고
    • Toward the optimal preconditioned eigensolver: Locally optimal block preconditioned conjugate gradient method
    • A.V. Knyazev Toward the optimal preconditioned eigensolver: locally optimal block preconditioned conjugate gradient method SIAM J. Sci. Comput. 23 2001 517 541
    • (2001) SIAM J. Sci. Comput. , vol.23 , pp. 517-541
    • Knyazev, A.V.1
  • 44
    • 6144274322 scopus 로고    scopus 로고
    • Spin gaps in a frustrated Heisenberg model for CaV4O9
    • S.R. White Spin gaps in a frustrated Heisenberg model for CaV4O9 Phys. Rev. Lett. 77 1996 3633 3636
    • (1996) Phys. Rev. Lett. , vol.77 , pp. 3633-3636
    • White, S.R.1
  • 45
    • 0038805356 scopus 로고    scopus 로고
    • Incomplete cross approximation in the Mosaic-Skeleton method
    • E.E. Tyrtyshnikov Incomplete cross approximation in the Mosaic-Skeleton method Computing 64 2000 367 380
    • (2000) Computing , vol.64 , pp. 367-380
    • Tyrtyshnikov, E.E.1
  • 47
    • 78649684489 scopus 로고    scopus 로고
    • Fast revealing of mode ranks of tensor in canonical format
    • D.V. Savostyanov Fast revealing of mode ranks of tensor in canonical format Numer. Math. Theory Methods Appl. 2 2009 439 444
    • (2009) Numer. Math. Theory Methods Appl. , vol.2 , pp. 439-444
    • Savostyanov, D.V.1
  • 48
    • 33751276962 scopus 로고
    • Numerical renormalization-group study of low-lying eigenstates of the antiferromagnetic S = 1 Heisenberg chain
    • S.R. White, and D. Huse Numerical renormalization-group study of low-lying eigenstates of the antiferromagnetic S = 1 Heisenberg chain Phys. Rev. B 48 1993 3844 3852
    • (1993) Phys. Rev. B , vol.48 , pp. 3844-3852
    • White, S.R.1    Huse, D.2
  • 49
    • 84867288584 scopus 로고    scopus 로고
    • Low-rank explicit QTT representation of the Laplace operator and its inverse
    • V.A. Kazeev, and B.N. Khoromskij Low-rank explicit QTT representation of the Laplace operator and its inverse SIAM J. Matrix Anal. Appl. 33 2012 742 758
    • (2012) SIAM J. Matrix Anal. Appl. , vol.33 , pp. 742-758
    • Kazeev, V.A.1    Khoromskij, B.N.2
  • 50
    • 0037115856 scopus 로고    scopus 로고
    • Benchmark calculations on high-dimensional Hénon-Heiles potentials with the multi-configuration time dependent Hartree (MCTDH) method
    • M. Nest, and H.-D. Meyer Benchmark calculations on high-dimensional Hénon-Heiles potentials with the multi-configuration time dependent Hartree (MCTDH) method J. Chem. Phys. 117 2002 10499
    • (2002) J. Chem. Phys. , vol.117 , pp. 10499
    • Nest, M.1    Meyer, H.-D.2
  • 51
    • 18344373359 scopus 로고
    • Generalised meshes for quantum mechanical problems
    • D. Baye, and P.-H. Heenen Generalised meshes for quantum mechanical problems J. Phys. A: Math. Gen. 19 1986 2041 2059
    • (1986) J. Phys. A: Math. Gen. , vol.19 , pp. 2041-2059
    • Baye, D.1    Heenen, P.-H.2


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.