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Volumn 34, Issue 6, 2012, Pages

Fast solution of parabolic problems in the tensor train/quantized tensor train format with initial application to the Fokker-Planck equation

Author keywords

Density matrix renormalization group; Dumbbell model; Fokker Planck equation; Higher dimensions; Parabolic problems; QTT format; Tensor methods

Indexed keywords

LINEAR SYSTEMS; STATISTICAL MECHANICS; TENSORS;

EID: 84871569236     PISSN: 10648275     EISSN: 10957200     Source Type: Journal    
DOI: 10.1137/120864210     Document Type: Article
Times cited : (126)

References (46)
  • 1
    • 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-Newtonian Fluid Mechanics, 139 (2006), pp. 153-176
    • (2006) J. Non-Newtonian Fluid Mechanics , vol.139 , pp. 153-176
    • Ammar, A.1    Mokdad, B.2    Chinesta, F.3    Keunings, R.4
  • 2
    • 79952566134 scopus 로고    scopus 로고
    • Finite element approximation of kinetic dilute polymer models with microscopic cut-off
    • J. W. Barrett and E. Süli, Finite element approximation of kinetic dilute polymer models with microscopic cut-off, ESAIM Math. Model. Numer. Anal., 45 (2011), pp. 39-89
    • (2011) ESAIM Math. Model. Numer. Anal. , vol.45 , pp. 39-89
    • Barrett, J.W.1    Süli, E.2
  • 5
    • 80255141549 scopus 로고    scopus 로고
    • Convergence of a greedy algorithm for highdimensional convex nonlinear problems
    • E. Cancés, V. Ehrlacher, and T. Leliévre, Convergence of a greedy algorithm for highdimensional convex nonlinear problems, Math. Models Methods Appl. Sci., 21 (2011), pp. 2433-2467
    • (2011) Math. Models Methods Appl. Sci. , vol.21 , pp. 2433-2467
    • Cancés, E.1    Ehrlacher, V.2    Leliévre, T.3
  • 6
    • 34250499792 scopus 로고
    • Analysis of individual differences in multidimensional scaling via n-way generalization of Eckart-Young decomposition
    • J. D. Caroll and J. J. Chang, Analysis of individual differences in multidimensional scaling via n-way generalization of Eckart-Young decomposition, Psychometrika, 35 (1970), pp. 283-319
    • (1970) Psychometrika , vol.35 , pp. 283-319
    • Caroll, J.D.1    Chang, J.J.2
  • 7
    • 0348146481 scopus 로고    scopus 로고
    • Simulation of dilute polymer solutions using a fokker-planck equation
    • C. Chauviére and A. Lozinski, Simulation of dilute polymer solutions using a Fokker-Planck equation, Comput. & Fluids, 33 (2004), pp. 687-696
    • (2004) Comput. & Fluids , vol.33 , pp. 687-696
    • Chauviére, C.1    Lozinski, A.2
  • 8
    • 55349142218 scopus 로고    scopus 로고
    • Tensor rank and the ill-posedness of the best low-rank approximation problem
    • V. de Silva and L.-H. Lim, Tensor rank and the ill-posedness of the best low-rank approximation problem, SIAM J. Matrix Anal. Appl., 30 (2008), pp. 1084-1127
    • (2008) SIAM J. Matrix Anal. Appl. , vol.30 , pp. 1084-1127
    • De Silva, V.1    Lim, L.-H.2
  • 10
    • 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), pp. A2718-A2739
    • (2012) SIAM J. Sci. Comput. , vol.34
    • Dolgov, S.V.1    Oseledets, I.V.2
  • 11
    • 84866540803 scopus 로고    scopus 로고
    • Greedy approximation of high-dimensional Ornstein-Uhlenbeck operators
    • L. E. Figueroa and E. Süli, Greedy approximation of high-dimensional Ornstein-Uhlenbeck operators, Found. Comput. Math., 12 (2012), pp. 573-623
    • (2012) Found. Comput. Math. , vol.12 , pp. 573-623
    • Figueroa, L.E.1    Süli, E.2
  • 12
    • 1442289165 scopus 로고
    • Die mittlere energie rotierender elektrischer dipole im strahlungsfeld
    • A. D. Fokker, Die mittlere Energie rotierender elektrischer Dipole im Strahlungsfeld, Ann. Phys., 348 (1914), pp. 810-820
    • (1914) Ann. Phys. , vol.348 , pp. 810-820
    • Fokker, A.D.1
  • 13
    • 80053908831 scopus 로고    scopus 로고
    • Quantized-TT-Cayley transform for computing the dynamics and the spectrum of high-dimensional Hamiltonians
    • I. Gavrilyuk and B. Khoromskij, Quantized-TT-Cayley transform for computing the dynamics and the spectrum of high-dimensional Hamiltonians, Comput. Methods Appl. Math., 11 (2011), pp. 273-290
    • (2011) Comput. Methods Appl. Math. , vol.11 , pp. 273-290
    • Gavrilyuk, I.1    Khoromskij, B.2
  • 14
    • 44049109914 scopus 로고
    • A rigorous derivation of the chemical master equation
    • D. T. Gillespie, A rigorous derivation of the chemical master equation, Phys. A, 188 (1992), pp. 404-425
    • (1992) Phys. A , vol.188 , pp. 404-425
    • Gillespie, D.T.1
  • 15
    • 38649130189 scopus 로고    scopus 로고
    • A sparse grid space-time discretization scheme for parabolic problems
    • M. Griebel and D. Oeltz, A sparse grid space-time discretization scheme for parabolic problems, Computing, 81 (2007), pp. 1-34
    • (2007) Computing , vol.81
    • Griebel, M.1    Oeltz, D.2
  • 16
    • 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
  • 18
    • 85162707890 scopus 로고
    • The expression of a tensor or a polyadic as a sum of products
    • F. L. Hitchcock, The expression of a tensor or a polyadic as a sum of products, J. Math. Phys, 6 (1927), pp. 164-189
    • (1927) J. Math. Phys , vol.6 , pp. 164-189
    • Hitchcock, F.L.1
  • 19
    • 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
    • Holtz, S.1    Rohwedder, T.2    Schneider, R.3
  • 20
    • 84858291641 scopus 로고    scopus 로고
    • On manifolds of tensors of fixed TT-rank
    • S. Holtz, T. Rohwedder, and R. Schneider, On manifolds of tensors of fixed TT-rank, Numer. Math., 120 (2012), pp. 701-731
    • (2012) Numer. Math. , vol.120 , pp. 701-731
    • Holtz, S.1    Rohwedder, T.2    Schneider, R.3
  • 21
    • 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
  • 22
    • 0010834538 scopus 로고
    • An alternating direction method for operator equations
    • R. B. Kellogg, An alternating direction method for operator equations, J. SIAM, 12 (1964), pp. 848-854
    • (1964) J. SIAM , vol.12 , pp. 848-854
    • Kellogg, R.B.1
  • 24
    • 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), pp. 257-280
    • (2011) Constr. Approx. , vol.34 , pp. 257-280
    • Khoromskij, B.N.1
  • 25
    • 83655192088 scopus 로고    scopus 로고
    • Tensor-structured numerical methods in scientific computing: Survey on recent advances
    • B. N. Khoromskij, Tensor-structured numerical methods in scientific computing: Survey on recent advances, Chemometr. Intell. Lab. Syst., 110 (2012), pp. 1-19
    • (2012) Chemometr. Intell. Lab. Syst. , vol.110
    • Khoromskij, B.N.1
  • 26
    • 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), pp. 3002-3026
    • (2009) SIAM J. Sci. Comput. , vol.31 , pp. 3002-3026
    • Khoromskij, B.N.1    Khoromskaia, V.2
  • 27
    • 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
  • 29
    • 68649096448 scopus 로고    scopus 로고
    • Tensor decompositions and applications
    • T. G. Kolda and B. W. Bader, Tensor decompositions and applications, SIAM Rev., 51 (2009), pp. 455-500
    • (2009) SIAM Rev. , vol.51 , pp. 455-500
    • Kolda, T.G.1    Bader, B.W.2
  • 31
    • 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), pp. 621-651
    • (2009) Constr. Approx. , vol.30 , pp. 621-651
    • Le Bris, C.1    Leliévre, T.2    Maday, Y.3
  • 32
    • 0042657876 scopus 로고    scopus 로고
    • A fast solver for fokker-planck equation applied to viscoelastic flows calculations: 2D FENE model
    • A. Lozinski and C. Chauviére, A fast solver for Fokker-Planck equation applied to viscoelastic flows calculations: 2D FENE model, J. Comput. Phys., 189 (2003), pp. 607-625
    • (2003) J. Comput. Phys. , vol.189 , pp. 607-625
    • Lozinski, A.1    Chauviére, C.2
  • 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), pp. 2130-2145
    • (2010) SIAM J. Matrix Anal. Appl. , vol.31 , pp. 2130-2145
    • Oseledets, I.V.1
  • 36
    • 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
  • 37
    • 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
  • 38
    • 70449525345 scopus 로고    scopus 로고
    • TT-cross approximation for multidimensional arrays
    • I. V. Oseledets and E. E. Tyrtyshnikov, TT-cross approximation for multidimensional arrays, Linear Algebra Appl., 432 (2010), pp. 70-88
    • (2010) Linear Algebra Appl. , vol.432 , pp. 70-88
    • Oseledets, I.V.1    Tyrtyshnikov, E.E.2
  • 39
    • 4244095121 scopus 로고
    • Thermodynamic limit of density matrix renormalization
    • S. Östlund and S. Rommer, Thermodynamic limit of density matrix renormalization, Phys. Rev. Lett., 75 (1995), pp. 3537-3540
    • (1995) Phys. Rev. Lett. , vol.75 , pp. 3537-3540
    • Östlund, S.1    Rommer, S.2
  • 41
    • 0022661834 scopus 로고
    • The exponential accuracy of fourier and chebychev differencing methods
    • E. Tadmor, The exponential accuracy of Fourier and Chebychev differencing methods, SIAM J. Numer. Anal., 23 (1986), pp. 1-23
    • (1986) SIAM J. Numer. Anal. , vol.23 , pp. 1-23
    • Tadmor, E.1
  • 42
    • 84868034256 scopus 로고    scopus 로고
    • Cambridge University Press, Cambridge, UK
    • V. Temlyakov, Greedy Approximation, Cambridge University Press, Cambridge, UK, 2011
    • (2011) Greedy Approximation
    • Temlyakov, V.1
  • 43
    • 12444298998 scopus 로고    scopus 로고
    • A QMC approach for high dimensional fokker-planck equations modelling polymeric liquids
    • G. Venkiteswaran and M. Junk, A QMC approach for high dimensional Fokker-Planck equations modelling polymeric liquids, Math. Comput. Simul., 68 (2005), pp. 43-56
    • (2005) Math. Comput. Simul. , vol.68 , pp. 43-56
    • Venkiteswaran, G.1    Junk, M.2
  • 45
    • 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), pp. 10345-10356
    • (1993) Phys. Rev. B , vol.48 , pp. 10345-10356
    • White, S.R.1
  • 46
    • 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), pp. 3844-3852.
    • (1993) Phys. Rev. B , vol.48 , pp. 3844-3852
    • White, S.R.1    Huse, D.2


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