메뉴 건너뛰기




Volumn 11, Issue 3, 2011, Pages 273-290

Quantized-TT-cayley transform for computing the dynamics and the spectrum of high-dimensional hamiltonians

Author keywords

Cayley transform; High dimensional problems; Matrix valued functions; Model reduction; Molecular dynamics; Quantized representation of vectors; Rank structured tensor approximation

Indexed keywords


EID: 80053908831     PISSN: 16094840     EISSN: 16099389     Source Type: Journal    
DOI: 10.2478/cmam-2011-0015     Document Type: Article
Times cited : (18)

References (44)
  • 2
    • 84963197428 scopus 로고
    • A method for solving initial value problems for linear difierential equations in Hilbert space based on the Cayley transform
    • D.Z.Arov and I.P. Gavrilyuk, A method for solving initial value problems for linear difierential equations in Hilbert space based on the Cayley transform, Numer. Funct. Anal. Optimization 14 (1993), no. 5-6, pp. 456-473.
    • (1993) Numer. Funct. Anal. Optimization , vol.14 , Issue.5-6 , pp. 456-473
    • Arov, D.Z.1    Gavrilyuk, I.P.2
  • 4
    • 0033636381 scopus 로고    scopus 로고
    • The multiconfiguration time-dependent Hartree (MCTDH) method: A highly eifcient algorithm for propagating wavepackets
    • M.H. Beck, A. Jäckle, G.A. Worth, and H.-D. Meyer, The multiconfiguration time-dependent Hartree (MCTDH) method: A highly eifcient algorithm for propagating wavepackets, Phys. Reports 324 (2000), pp. 1-105.
    • (2000) Phys. Reports , vol.324 , pp. 1-105
    • Beck, M.H.1    Jäckle, A.2    Worth, G.A.3    Meyer, H.-D.4
  • 6
    • 80053913107 scopus 로고    scopus 로고
    • Super exponentially convergent approximation to the solution of the Schrdinger equation in abstract setting
    • I.P. Gavrilyuk, Super exponentially convergent approximation to the solution of the Schrdinger equation in abstract setting, CMAM, 10 (2010), no. 4, pp. 345-358.
    • (2010) CMAM , vol.10 , Issue.4 , pp. 345-358
    • Gavrilyuk, I.P.1
  • 7
    • 14944359407 scopus 로고    scopus 로고
    • Data-sparse approximation to a class of operator-valued functions
    • DOI 10.1090/S0025-5718-04-01703-X, PII S002557180401703X
    • I.P. Gavrilyuk, W. Hackbusch, and B.N. Khoromskij, Data-sparse approximation to a class of operatorvalued functions, Math. Comp. 74 (2005), pp. 681-708. (Pubitemid 40367036)
    • (2005) Mathematics of Computation , vol.74 , Issue.250 , pp. 681-708
    • Gavrilyuk, I.P.1    Hackbusch, W.2    Khoromskij, B.N.3
  • 8
    • 17144410769 scopus 로고    scopus 로고
    • Hierarchical tensor-product approximation to the inverse and related operators for high-dimensional elliptic problems
    • DOI 10.1007/s00607-004-0086-y
    • 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. (Pubitemid 40512277)
    • (2005) Computing (Vienna/New York) , vol.74 , Issue.2 , pp. 131-157
    • Gavrilyuk, I.P.1    Hackbusch, W.2    Khoromskij, B.N.3
  • 9
    • 0043139068 scopus 로고    scopus 로고
    • Exact and approximate solutions of some operator equations based on the Cayley transform
    • PII S0024379598100502
    • I.P. Gavrilyuk and V.L. Makarov, Exact and approximate solutions of some operator equations based on the Cayley transform, Linear Algebra and its Applications, 282(1998), pp. 97-121. (Pubitemid 128399746)
    • (1998) Linear Algebra and Its Applications , vol.282 , Issue.1-3 , pp. 97-121
    • Gavrilyuk, I.P.1    Makarov, V.L.2
  • 10
    • 33749034221 scopus 로고    scopus 로고
    • Exponentially convergent algorithms for the operator exponential with applications to inhomogeneous problems in Banach spaces
    • DOI 10.1137/040611045
    • I.P. Gavrilyuk and V.L. Makarov, Exponentially convergent algorithms for the operator exponential with applications to inhomogeneous problems in Banach spaces, SIAM Journal on Num. Anal., 43 (2005), no. 5 , pp. 2144-2171. (Pubitemid 44598676)
    • (2005) SIAM Journal on Numerical Analysis , vol.43 , Issue.5 , pp. 2144-2171
    • Gavrilyuk, I.P.1    Makarov, V.L.2
  • 13
    • 34547545863 scopus 로고    scopus 로고
    • Sparse grids for the Schrödinger equation
    • M. Griebel and J. Hamaekers, Sparse grids for the Schrödinger equation, M2AN, 41 (2007), pp. 215-247.
    • (2007) M2AN , vol.41 , pp. 215-247
    • Griebel, M.1    Hamaekers, J.2
  • 15
    • 38649130189 scopus 로고    scopus 로고
    • A sparse grid space-time discretization scheme for parabolic problems
    • DOI 10.1007/s00607-007-0241-3
    • M. Griebel, and D. Oeltz, A Sparse Grid Space-Time Discretization Scheme for Parabolic Problems, Computing, 81 (2007), no. 1, pp. 1-34. (Pubitemid 351168840)
    • (2007) Computing (Vienna/New York) , vol.81 , Issue.1 , pp. 1-34
    • Griebel, M.1    Oeltz, D.2
  • 16
    • 29144513721 scopus 로고    scopus 로고
    • Low-rank Kronecker-product approximation to multi-dimensional nonlocal operators. Part I. Separable approximation of multi-variate functions
    • DOI 10.1007/s00607-005-0144-0
    • W. Hackbusch and B.N. Khoromskij: Low-rank Kronecker product approximation to multi-dimensional nonlocal operators. Part I. Separable approximation of multi-variate functions, Computing, 76 (2006), pp. 177-202. (Pubitemid 41813085)
    • (2006) Computing (Vienna/New York) , vol.76 , Issue.3-4 , pp. 177-202
    • Hackbusch, W.1    Khoromskij, B.N.2
  • 17
  • 21
    • 34447532966 scopus 로고    scopus 로고
    • Structured rank-(r1;rd) decomposition of function-related tensors in rd comp
    • B.N. Khoromskij, Structured Rank-(r1;rd) Decomposition of Function-related Tensors in Rd, Comp. Meth. in Applied Math., 6 (2006), no. 2, pp. 194-220.
    • (2006) Meth. in Applied Math. , vol.6 , Issue.2 , pp. 194-220
    • Khoromskij, B.N.1
  • 22
    • 77955278344 scopus 로고    scopus 로고
    • Fast and Accurate Tensor Approximation of a Multivariate Convolution with Linear Scaling in Dimension
    • B.N. Khoromskij, Fast and Accurate Tensor Approximation of a Multivariate Convolution with Linear Scaling in Dimension, J. of Comp. Appl. Math., 234 (2010), pp. 3122-3139.
    • (2010) J. of Comp. Appl. Math. , vol.234 , pp. 3122-3139
    • Khoromskij, B.N.1
  • 23
    • 70449726943 scopus 로고    scopus 로고
    • Tensor-structured preconditioners and approximate inverse of elliptic operators in Rd
    • B.N. Khoromskij, Tensor-Structured Preconditioners and Approximate Inverse of Elliptic Operators in Rd, J. Constructive Approx, 30 (2009), pp. 599-620.
    • (2009) J. Constructive Approx , vol.30 , pp. 599-620
    • Khoromskij, B.N.1
  • 24
    • 79960994573 scopus 로고    scopus 로고
    • O(d logN)-Quantics Approximation of N-d Tensors in High-Dimensional Numerical Modeling
    • Preprint 55/2009 MPI MIS, Leipzig 2009
    • B.N. Khoromskij, O(d logN)-Quantics Approximation of N-d Tensors in High-Dimensional Numerical Modeling, J. Constr. Approx. 2011, Preprint 55/2009 MPI MIS, Leipzig 2009.
    • (2011) J. Constr. Approx.
    • Khoromskij, B.N.1
  • 25
    • 80053895144 scopus 로고    scopus 로고
    • Tensors-structured numerical methods in scientific computing: Survey on recent advances
    • Preprint 21/2010, MPI MiS Leipzig 2010
    • B.N. Khoromskij, Tensors-structured Numerical Methods in Scientific Computing: Survey on Recent Advances, Chemometrics and Intelligent Laboratory Systems, 2011, Preprint 21/2010, MPI MiS Leipzig 2010.
    • (2011) Chemometrics and Intelligent Laboratory Systems
    • Khoromskij, B.N.1
  • 26
    • 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. on Sci. Comp., 33 (2011), no. 1, pp. 45-65.
    • (2011) SIAM J. on Sci. Comp. , vol.33 , Issue.1 , pp. 45-65
    • Khoromskij, B.N.1    Khoromskaia, V.2    Flad, H.-J.3
  • 27
    • 79960448781 scopus 로고    scopus 로고
    • Quantics-TT approximation of elliptic solution operators in higher dimensions
    • B.N. Khoromskij and I. Oseledets, Quantics-TT approximation of elliptic solution operators in higher dimensions, Russ. J. Numer. Anal. Math. Modelling, 26 (2011), no. 3, pp. 303-322.
    • (2011) Russ. J. Numer. Anal. Math. Modelling , vol.26 , Issue.3 , pp. 303-322
    • Khoromskij, B.N.1    Oseledets, I.2
  • 29
    • 68649096448 scopus 로고    scopus 로고
    • Tensor decompositions and applications
    • T.G. Kolda and B.W. Bader, Tensor Decompositions and Applications, SIAM Review, 51 (2009), no. 3, pp. 455-500.
    • (2009) SIAM Review , vol.51 , Issue.3 , pp. 455-500
    • Kolda, T.G.1    Bader, B.W.2
  • 30
    • 19844366465 scopus 로고    scopus 로고
    • Computational chemistry from the perspective of numerical analysis
    • C. Le Bris, Computational chemistry from the perspective of numerical analysis, Acta Numerica (2005), pp. 363-444.
    • (2005) Acta Numerica , vol.444 , pp. 363
    • Le Bris, C.1
  • 34
    • 77956042755 scopus 로고    scopus 로고
    • Approximation of 2d × 2d matrices using tensor decomposition
    • I.V. Oseledets, Approximation of 2d × 2d matrices using tensor decomposition, SIAM J. Matrix Anal. Appl., 31 (2010), no. 4, pp. 2130-2145.
    • (2010) SIAM J. Matrix Anal. Appl. , vol.31 , Issue.4 , pp. 2130-2145
    • Oseledets, I.V.1
  • 35
    • 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. Comp., 31 (2009), pp. 3744-3759.
    • (2009) SIAM J. Sci. Comp. , vol.31 , pp. 3744-3759
    • Oseledets, I.V.1    Tyrtyshnikov, E.E.2
  • 36
    • 1342326246 scopus 로고    scopus 로고
    • Numerical solution of parabolic equations in high dimensions
    • T. Petersdorfi, and C. Schwab, Numerical solution of parabolic equations in high dimensions, Math Model Numer Anal 38 (2004), pp. 93-128.
    • (2004) Math Model Numer Anal , vol.38 , pp. 93-128
    • Petersdorfi, T.1    Schwab, C.2
  • 38
    • 0242370956 scopus 로고    scopus 로고
    • Sparse finite elements for elliptic problems with stochastic loading
    • C. Schwab and R. Todor, Sparse finite elements for elliptic problems with stochastic loading, Numer. Math., 95 (2003), no. 4, pp. 707-713.
    • (2003) Numer. Math. , vol.95 , Issue.4 , pp. 707-713
    • Schwab, C.1    Todor, R.2
  • 39
    • 0002522806 scopus 로고    scopus 로고
    • When are quasi-monte carlo algorithms efficient for high dimensional integrals?
    • I. Sloan, and H. Wozniakowski, When Are Quasi-Monte Carlo Algorithms Eifcient for High Dimensional Integrals, J. of Complexity, 14 (1998), no. 1, pp. 1-33. (Pubitemid 128421301)
    • (1998) Journal of Complexity , vol.14 , Issue.1 , pp. 1-33
    • Sloan, I.H.1    Wozniakowski, H.2
  • 41
    • 0004073954 scopus 로고
    • American Mathematical Society, New York
    • G. Szegö, Orthogonal Polynomials, American Mathematical Society, New York, 1959.
    • (1959) Orthogonal Polynomials
    • Szegö, G.1
  • 42
    • 0242425255 scopus 로고    scopus 로고
    • Eifcient classical simulation of slightly entanhgled quantum computations
    • G. Vidal, Eifcient classical simulation of slightly entanhgled quantum computations, Phys. Rev. Lett. 91 (2003), no. 14, 147902-1 147902-4.
    • (2003) Phys. Rev. Lett. , vol.91 , Issue.14 , pp. 1479021-1479024
    • Vidal, G.1
  • 43
    • 0042767657 scopus 로고    scopus 로고
    • Multilayer formulation of the multiconfiguration time-dependent Hartree theory
    • H.Wang, and M. Thoss, Multilayer formulation of the multiconfiguration time-dependent Hartree theory, J. Chem. Phys. 119 (2003), pp. 1289-1299.
    • (2003) J. Chem. Phys. , vol.119 , pp. 1289-1299
    • Wang, H.1    Thoss, M.2
  • 44
    • 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), no. 14, pp. 10345-10356.
    • (1993) Phys. Rev. B , vol.48 , Issue.14 , pp. 10345-10356
    • White, S.R.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.