Time integration of tensor trains

SIAM Journal on Numerical Analysis

Volumn 53, Issue 2, 2015, Pages 917-941

  Lubich, Christian ^a   Oseledets, Ivan V ^b,c   Vandereycken, Bart ^d  

^a UNIVERSITY OF TÜBINGEN   (Germany)

^b SKOLKOVO INSTITUTE OF SCIENCE AND TECHNOLOGY   (Russian Federation)

^c INSTITUTE OF NUMERICAL MATHEMATICS   (Russian Federation)

^d PRINCETON UNIVERSITY   (United States)

Author keywords

Low rank approximation; Matrix product state; Splitting integrator; Tensor differential equations; Tensor train; Time varying tensors

Indexed keywords

APPROXIMATION THEORY; DIFFERENTIAL EQUATIONS; ITERATIVE METHODS; MATRIX ALGEBRA; MOLECULAR DYNAMICS;

LOW RANK APPROXIMATIONS; MATRIX PRODUCT STATE; SPLITTING INTEGRATOR; TENSOR TRAINS; TIME VARYING;

TENSORS;

EID: 84929448608     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/140976546     Document Type: Article

References (32)

1. P.-A. ABSIL, R. MAHONY, AND R. SEPULCHRE, Optimization Algorithms on Matrix Manifolds, Princeton University Press, Princeton, NJ, 2008.
2. M. H. BECK, A. JÄCKLE, G. A. WORTH, AND H.-D. MEYER, The multiconfiguration time-dependent Hartree method: A highly efficient algorithm for propagating wavepackets, Phys. Rep., 324 (2000), pp. 1-105.
3. D. T. COLBERT AND W. H. MILLER, A novel discrete variable representation for quantum mechanical reactive scattering via the S-matrix Kohn method, J. Chem. Phys., 96 (1992), pp. 1982-1991.
4. W. HACKBUSCH, Tensor Spaces and Numerical Tensor Calculus, Springer, Berlin, 2012.
5. W. HACKBUSCH, B. KHOROMSKIJ, AND E. TYRTYSHNIKOV, Approximate iterations for structured matrices, Numer. Math., 109 (2008), pp. 365-383.
6. J. HAEGEMAN, T. J. OSBORNE, AND F. VERSTRAETE, Post-matrix product state methods: To tangent space and beyond, Phys. Rev. B, 88 (2013), 075133.
7. E. HAIRER, C. LUBICH, AND G. WANNER, Geometric Numerical Integration, 2nd ed., Springer- Verlag, Berlin, 2006.
8. M. HOCHBRUCK AND C. LUBICH, On Krylov subspace approximations to the matrix exponential operator, SIAM J. Numer. Anal., 34 (1997), pp. 1911-1925.
9. M. HOCHBRUCK AND A. OSTERMANN, Exponential integrators, Acta Numer., 19 (2010), pp. 209-286.
10. 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.
11. S. HOLTZ, T. ROHWEDDER, AND R. SCHNEIDER, On manifolds of tensors of fixed TT-rank, Numer. Math., 120 (2012), pp. 701-731.
12. 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.
13. 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.
14. O. KOCH AND C. LUBICH, Dynamical low-rank approximation, SIAM J. Matrix Anal. Appl., 29 (2007), pp. 434-454.
15. T. G. KOLDA AND B. W. BADER, Tensor decompositions and applications, SIAM Rev., 51 (2009), pp. 455-500.
16. P. KRAMER AND M. SARACENO, Geometry of the Time-Dependent Variational Principle in Quantum Mechanics, Lecture Notes in Phys. 140, Springer-Verlag, Berlin, New York, 1981.
17. C. LUBICH, From Quantum to Classical Molecular Dynamics: Reduced Models and Numerical Analysis, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2008.
18. C. LUBICH AND I. OSELEDETS, A projector-splitting integrator for dynamical low-rank approximation, BIT, 54 (2014), pp. 171-188.
19. C. LUBICH, T. ROHWEDDER, R. SCHNEIDER, AND B. VANDEREYCKEN, Dynamical approximation by hierarchical Tucker and tensor-train tensors, SIAM J. Matrix Anal. Appl., 34 (2013), pp. 470-494.
20. U. MANTHE, H.-D. MEYER, AND L. S. CEDERBAUM, Wave-packet dynamics within the multiconfiguration Hartree framework: General aspects and application to NOCl, J. Chem. Phys., 97 (1992), pp. 3199-3213.
21. H.-D. MEYER, F. GATTI, AND G. A. WORTH, EDS., Multidimensional Quantum Dynamics: MCTDH Theory and Applications, Wiley-VCH, Weinheim, Germany, 2009.
22. H.-D. MEYER AND G. A. WORTH, Quantum molecular dynamics: Propagating wavepackets and density operators using the multiconfiguration time-dependent Hartree (MCTDH) method, Theor. Chem. Acc., 109 (2003), pp. 251-267.
23. M. NEST AND H.-D. MEYER, Benchmark calculations on high-dimensional Henon-Heiles potentials with the multi-configuration time dependent Hartree (MCTDH) method, J. Chem. Phys., 117 (2002), pp. 10499-10505.
24. d matrices using tensor decomposition, SIAM J. Matrix Anal. Appl., 31 (2010), pp. 2130-2145.
25. I. V. OSELEDETS, Tensor-train decomposition, SIAM J. Sci. Comput., 33 (2011), pp. 2295-2317.
26. I. V. OSELEDETS AND S. V. DOLGOV, Solution of linear systems and matrix inversion in the TT-format, SIAM J. Sci. Comput., 34 (2012), pp. A2718-A2739.
27. 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.
28. 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.
29. U. SCHOLLWÖCK, The density-matrix renormalization group in the age of matrix product states, Ann. Phys., 326 (2011), pp. 96-192.
30. R. B. SIDJE, Expokit: A software package for computing matrix exponentials, ACM Trans. Math. Software, 24 (1998), pp. 130-156.
31. A. USCHMAJEW AND B. VANDEREYCKEN, The geometry of algorithms using hierarchical tensors, Linear Algebra Appl., 439 (2013), pp. 133-166.
32. G. A. WORTH, M. H. BECK, A. JÄCKLE, AND H.-D. MEYER, The MCTDH Package, Version 8.2, 2000 [H.-D. Meyer, Version 8.3, 2002; Version 8.4, 2007], http://mctdh.uni-hd.de.