메뉴 건너뛰기




Volumn 36, Issue 5, 2014, Pages A2346-A2368

Low-rank tensor methods with subspace correction for symmetric eigenvalue problems

Author keywords

ALS; DMRG; High dimensional eigenvalue problems; LOBPCG; Low rank tensor methods; Tensor train format; Trace minimization

Indexed keywords

EIGENVALUES AND EIGENFUNCTIONS; TENSORS;

EID: 84911401665     PISSN: 10648275     EISSN: 10957197     Source Type: Journal    
DOI: 10.1137/130949919     Document Type: Article
Times cited : (60)

References (43)
  • 2
    • 0036494136 scopus 로고    scopus 로고
    • A Grassmann-Rayleigh quotient iteration for computing invariant subspaces
    • P. A. Absil, R. Mahony, R. Sepulchre, and P. Van Dooren, A Grassmann-Rayleigh quotient iteration for computing invariant subspaces, SIAM Rev., 44 (2002), pp. 57-73.
    • (2002) SIAM Rev. , vol.44 , pp. 57-73
    • Absil, P.A.1    Mahony, R.2    Sepulchre, R.3    Van Dooren, P.4
  • 3
    • 18344373359 scopus 로고
    • Generalised meshes for quantum mechanical problems
    • D. Baye and P.-H. Heenen, Generalised meshes for quantum mechanical problems, J. Phys. A, 19 (1986), pp. 2041-2059.
    • (1986) J. Phys. A , vol.19 , pp. 2041-2059
    • Baye, D.1    Heenen, P.-H.2
  • 4
    • 0344845742 scopus 로고    scopus 로고
    • A subspace preconditioning algorithm for eigenvector/eigenvalue computation
    • J. H. Bramble, J. E. Pasciak, and A. V. Knyazev, A subspace preconditioning algorithm for eigenvector/eigenvalue computation, Adv. Comput. Math., 6 (1996), pp. 159-189.
    • (1996) Adv. Comput. Math. , vol.6 , pp. 159-189
    • Bramble, J.H.1    Pasciak, J.E.2    Knyazev, A.V.3
  • 5
    • 84871571335 scopus 로고    scopus 로고
    • Periodic Schrödinger operators with local defects and spectral pollution
    • E. Cances, V. Ehrlacher, and Y. Maday, Periodic Schrödinger operators with local defects and spectral pollution, SIAM J. Numer. Anal., 50 (2012), pp. 3016-3035.
    • (2012) SIAM J. Numer. Anal. , vol.50 , pp. 3016-3035
    • Cances, E.1    Ehrlacher, V.2    Maday, Y.3
  • 6
    • 84894609844 scopus 로고    scopus 로고
    • Computation of extreme eigenvalues in higher dimensions using block tensor train format
    • S. V. Dolgov, B. N. Khoromskij, I. V. Oseledets, and D. V. Savostyanov, Computation of extreme eigenvalues in higher dimensions using block tensor train format, Comput. Phys. Comm., 185 (2014), pp. 1207-1216.
    • (2014) Comput. Phys. Comm. , vol.185 , pp. 1207-1216
    • Dolgov, S.V.1    Khoromskij, B.N.2    Oseledets, I.V.3    Savostyanov, D.V.4
  • 7
    • 84857675961 scopus 로고    scopus 로고
    • Low-rank tensor structure of solutions to elliptic problems with jumping coefficients
    • S. V. Dolgov, B. N. Khoromskij, I. V. Oseledets, and E. E. Tyrtyshnikov, Low-rank tensor structure of solutions to elliptic problems with jumping coefficients, J. Comput. Math., 30 (2012), pp. 14-23.
    • (2012) J. Comput. Math. , vol.30 , pp. 14-23
    • Dolgov, S.V.1    Khoromskij, B.N.2    Oseledets, I.V.3    Tyrtyshnikov, E.E.4
  • 11
    • 0032216898 scopus 로고    scopus 로고
    • The geometry of algorithms with orthogonality constraints
    • A. Edelman, T. A. Arias, and S. T. Smith, The geometry of algorithms with orthogonality constraints, SIAM J. Matrix Anal. Appl., 20 (1998), pp. 303-353.
    • (1998) SIAM J. Matrix Anal. Appl. , vol.20 , pp. 303-353
    • Edelman, A.1    Arias, T.A.2    Smith, S.T.3
  • 12
    • 77953845148 scopus 로고    scopus 로고
    • Computing semiclassical quantum dynamics with Hagedorn wavepackets
    • E. Faou, V. Gradinaru, and C. Lubich, Computing semiclassical quantum dynamics with Hagedorn wavepackets, SIAM J. Sci. Comput., 31 (2009), pp. 3027-3041.
    • (2009) SIAM J. Sci. Comput. , vol.31 , pp. 3027-3041
    • Faou, E.1    Gradinaru, V.2    Lubich, C.3
  • 13
    • 3042756404 scopus 로고    scopus 로고
    • Existence and computation of low Kronecker-rank approximations for large linear systems of tensor product structure
    • L. Grasedyck, Existence and computation of low Kronecker-rank approximations for large linear systems of tensor product structure, Computing, 72 (2004), pp. 247-265.
    • (2004) Computing , vol.72 , pp. 247-265
    • Grasedyck, L.1
  • 14
    • 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), pp. 2029-2054.
    • (2010) SIAM J. Matrix Anal. Appl. , vol.31 , pp. 2029-2054
    • Grasedyck, L.1
  • 19
    • 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), pp. 706-722.
    • (2009) J. Fourier Anal. Appl. , vol.15 , pp. 706-722
    • Hackbusch, W.1    Kühn, S.2
  • 20
    • 11044224123 scopus 로고    scopus 로고
    • Multiresolution quantum chemistry: Basic theory and initial applications
    • R. J. Harrison, G. I. Fann, T. Yanai, Z. Gan, and G. Beylkin, Multiresolution quantum chemistry: Basic theory and initial applications, J. Chem. Phys., 121 (2004), pp. 11587-11598.
    • (2004) J. Chem. Phys. , vol.121 , pp. 11587-11598
    • Harrison, R.J.1    Fann, G.I.2    Yanai, T.3    Gan, Z.4    Beylkin, G.5
  • 21
    • 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), pp. A683-A713.
    • (2012) SIAM J. Sci. Comput. , vol.34 , pp. A683-A713
    • Holtz, S.1    Rohwedder, T.2    Schneider, R.3
  • 22
    • 84900564375 scopus 로고    scopus 로고
    • Subspace iteration methods in terms of matrix product states
    • T. Huckle and K. Waldherr, Subspace iteration methods in terms of matrix product states, PAMM Proc. Appl. Math. Mech., 12 (2012), pp. 641-642.
    • (2012) PAMM Proc. Appl. Math. Mech. , vol.12 , pp. 641-642
    • Huckle, T.1    Waldherr, K.2
  • 23
    • 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), pp. 742-758.
    • (2012) SIAM J. Matrix Anal. Appl. , vol.33 , pp. 742-758
    • Kazeev, V.A.1    Khoromskij, B.N.2
  • 24
    • 84898646167 scopus 로고    scopus 로고
    • Black-box Hartree-Fock solver by tensor numerical methods
    • V. Khoromskaia, Black-box Hartree-Fock solver by tensor numerical methods, Comput. Methods Appl. Math., 14 (2014), pp. 89-111.
    • (2014) Comput. Methods Appl. Math. , vol.14 , pp. 89-111
    • Khoromskaia, V.1
  • 25
    • 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), pp. 45-65.
    • (2011) SIAM J. Sci. Comput. , vol.33 , pp. 45-65
    • Khoromskij, B.N.1    Khoromskaia, V.2    Flad, H.-J.3
  • 27
    • 80053917258 scopus 로고    scopus 로고
    • Quantics-TT collocation approximation of parameterdependent and stochastic elliptic PDEs
    • B. N. Khoromskij and I. V. Oseledets, Quantics-TT collocation approximation of parameterdependent and stochastic elliptic PDEs, Comput. Methods Appl. Math., 10 (2010), pp. 376-394.
    • (2010) Comput. Methods Appl. Math. , vol.10 , pp. 376-394
    • Khoromskij, B.N.1    Oseledets, I.V.2
  • 28
    • 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), pp. 517-541.
    • (2001) SIAM J. Sci. Comput. , vol.23 , pp. 517-541
    • Knyazev, A.V.1
  • 29
    • 80053926097 scopus 로고    scopus 로고
    • Preconditioned low-rank methods for high-dimensional elliptic PDE eigenvalue problems
    • D. Kressner and C. Tobler, Preconditioned low-rank methods for high-dimensional elliptic PDE eigenvalue problems, Comput. Methods Appl. Math., 11 (2011), pp. 363-381.
    • (2011) Comput. Methods Appl. Math. , vol.11 , pp. 363-381
    • Kressner, D.1    Tobler, C.2
  • 30
    • 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, Russian J. Numer. Anal. Math. Modelling, 26 (2011), pp. 465-489.
    • (2011) Russian J. Numer. Anal. Math. Modelling , vol.26 , pp. 465-489
    • Lebedeva, O.S.1
  • 31
    • 0002053823 scopus 로고
    • The multiconfigurational time-dependent Hartree approach
    • H.-D. Meyer, U. Manthe, and L.-S. Cederbaum, The multiconfigurational time-dependent Hartree approach, Chem. Phys. Lett., 165 (1990), pp. 73-78.
    • (1990) Chem. Phys. Lett. , vol.165 , pp. 73-78
    • Meyer, H.-D.1    Manthe, U.2    Cederbaum, L.-S.3
  • 32
    • 0036003403 scopus 로고    scopus 로고
    • A geometric theory for preconditioned inverse iteration applied to a subspace
    • K. Neymeyr, A geometric theory for preconditioned inverse iteration applied to a subspace, Math. Comp., 71 (2002), pp. 197-216.
    • (2002) Math. Comp. , vol.71 , pp. 197-216
    • Neymeyr, K.1
  • 33
    • 80053896203 scopus 로고    scopus 로고
    • Tensor-train decomposition
    • I. V. Oseledets, Tensor-train decomposition, SIAM J. Sci. Comput., 33 (2011), pp. 2295-2317.
    • (2011) SIAM J. Sci. Comput. , vol.33 , pp. 2295-2317
    • Oseledets, I.V.1
  • 34
    • 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), pp. 3744-3759.
    • (2009) SIAM J. Sci. Comput. , vol.31 , pp. 3744-3759
    • Oseledets, I.V.1    Tyrtyshnikov, E.E.2
  • 35
    • 84856895584 scopus 로고    scopus 로고
    • Variational numerical renormalization group: Bridging the gap between NRG and density matrix renormalization group
    • I. Pi?zorn and F. Verstraete, Variational numerical renormalization group: Bridging the gap between NRG and density matrix renormalization group, Phys. Rev. Lett., 108 (2012), 067202.
    • (2012) Phys. Rev. Lett. , vol.108 , pp. 067202
    • Pizorn, I.1    Verstraete, F.2
  • 36
    • 0033691032 scopus 로고    scopus 로고
    • A numerical study on the performance of the multiconfigurational time-dependent Hartree method for density operators
    • A. Raab and H.-D. Meyer, A numerical study on the performance of the multiconfigurational time-dependent Hartree method for density operators, J. Chem. Phys., 112 (2000), pp. 10718-10729.
    • (2000) J. Chem. Phys. , vol.112 , pp. 10718-10729
    • Raab, A.1    Meyer, H.-D.2
  • 37
    • 84880562542 scopus 로고    scopus 로고
    • On local convergence of alternating schemes for optimization of convex problems in the tensor train format
    • T. Rohwedder and A. Uschmajew, On local convergence of alternating schemes for optimization of convex problems in the tensor train format, SIAM J. Numer. Anal., 51 (2013), pp. 1134-1162.
    • (2013) SIAM J. Numer. Anal. , vol.51 , pp. 1134-1162
    • Rohwedder, T.1    Uschmajew, A.2
  • 38
    • 0034319898 scopus 로고    scopus 로고
    • The trace minimization method for the symmetric generalized eigenvalue problem
    • A. H. Sameh and Z. Tong, The trace minimization method for the symmetric generalized eigenvalue problem, J. Comput. Appl. Math., 123 (2000), pp. 155-175.
    • (2000) J. Comput. Appl. Math. , vol.123 , pp. 155-175
    • Sameh, A.H.1    Tong, Z.2
  • 39
    • 0009148594 scopus 로고
    • A trace minimization algorithm for the generalized eigenvalue problem
    • A. H. Sameh and J. A. Wisniewski, A trace minimization algorithm for the generalized eigenvalue problem, SIAM J. Numer. Anal., 19 (1982), pp. 1243-1259.
    • (1982) SIAM J. Numer. Anal. , vol.19 , pp. 1243-1259
    • Sameh, A.H.1    Wisniewski, J.A.2
  • 40
    • 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), pp. 96-192.
    • (2011) Ann. Physics , vol.326 , pp. 96-192
    • Schollwöck, U.1
  • 41
    • 72449185787 scopus 로고    scopus 로고
    • Variational matrix-product-state approach to quantum impurity models
    • A. Weichselbaum, F. Verstraete, U. Schollwöck, J. I. Cirac, and J. von Delft, Variational matrix-product-state approach to quantum impurity models, Phys. Rev. B, 80 (2009), 165117.
    • (2009) Phys. Rev. B , vol.80 , pp. 165117
    • Weichselbaum, A.1    Verstraete, F.2    Schollwöck, U.3    Cirac, J.I.4
  • 42
    • 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), pp. 2863-2866.
    • (1992) Phys. Rev. Lett. , vol.69 , pp. 2863-2866
    • White, S.R.1
  • 43
    • 29644437321 scopus 로고    scopus 로고
    • Density matrix renormalization group algorithms with a single center site
    • S. R. White, Density matrix renormalization group algorithms with a single center site, Phys. Rev. B, 72 (2005), 180403.
    • (2005) Phys. Rev. B , vol.72 , pp. 180403
    • White, S.R.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.