Startseite Mathematik Projection Methods for Dynamical Low-Rank Approximation of High-Dimensional Problems
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Projection Methods for Dynamical Low-Rank Approximation of High-Dimensional Problems

  • Emil Kieri und Bart Vandereycken EMAIL logo
Veröffentlicht/Copyright: 21. Juli 2018
Computational Methods in Applied Mathematics
Aus der Zeitschrift Computational Methods in Applied Mathematics Band 19 Heft 1

Abstract

We consider dynamical low-rank approximation on the manifold of fixed-rank matrices and tensor trains (also called matrix product states), and analyse projection methods for the time integration of such problems. First, under suitable approximability assumptions, we prove error estimates for the explicit Euler method equipped with quasi-optimal projections to the manifold. Then we discuss the possibilities and difficulties with higher-order explicit methods. In particular, we discuss ways for limiting rank growth in the increments, and robustness with respect to small singular values.

Keywords: Tensor Train; Low-Rank Approximation; Tensor Differential Equations; Projection Methods
MSC 2010: 58J35; 65L05; 65L06; 65L70

Funding statement: Bart Vandereycken was partly supported by SNF project 159856 entitled “Analyse numérique”.

A Tightness of Curvature Bound

We here show that the bound (1.4) is sharp in the sense that for any X, we can choose Y and Z such that the bound is attained. To keep the notation manageable, we restrict ourselves to the case of square matrices.

We consider the manifold ℳr of real N×N matrices of rank r. Let X∈ℳr be any matrix on the manifold, and construct its SVD X=U⁢S⁢VT such that U,V∈ℝN×r have orthonormal columns ui and vi, respectively, and S∈ℝr×r is a diagonal matrix with elements si⁢i=σi>0 in decreasing order. Next, choose Y∈ℳr such that Y=U⁢S⁢V~T, where also V~∈ℝN×r has orthonormal columns v~i, and vi=v~i for i=1,…,r-1. Then

Y-X=σr⁢ur⁢(v~r-vr)T,

and thereby

(A.1)∥Y-X∥2=tr⁡((Y-X)T⁢(Y-X))=2⁢σr2⁢(1-v~rT⁢vr).

Now take any Z∈ℝN×N. Since P⁢(X)⁢Z=U⁢UT⁢Z+Z⁢V⁢VT-U⁢UT⁢Z⁢V⁢VT (see, e.g., [18, equation (2.5)]), we get

(P⁢(Y)-P⁢(X))⁢Z=(I-U⁢UT)⁢Z⁢(V~⁢V~T-V⁢VT)=(I-U⁢UT)⁢Z⁢(v~r⁢v~rT-vr⁢vrT).

We choose Z=u~⁢v~rT, where u~ is normalised and orthogonal to the columns of U. Then ∥Z∥=1 and

∥(P⁢(Y)-P⁢(X))⁢Z∥2=∥u~⁢(v~rT-v~rT⁢vr⁢vrT)∥2=1-(v~rT⁢vr)2.

Since vr and v~r are normalised, we have |v~rT⁢vr|≤1. We choose v~r such that 0<v~rT⁢vr<1. Then

∥(P⁢(Y)-P⁢(X))⁢Z∥=1-(v~rT⁢vr)2≥1-v~rT⁢vr,

and by (A.1) and ∥Z∥=1, we get

∥(P⁢(Y)-P⁢(X))⁢Z∥≥12⁢σr⁢∥Y-X∥⁢∥Z∥,

which shows the claim that (1.4) is essentially a tight estimate.

Acknowledgements

The second author thanks Daniel Kressner for initial discussions about projected integrators in the context of low-rank matrices.

References

[1] P.-A. Absil and I. V. Oseledets, Low-rank retractions: A survey and new results, Comput. Optim. Appl. 62 (2015), no. 1, 5–29. 10.1007/s10589-014-9714-4Suche in Google Scholar

[2] M. H. Beck, A. Jäckle, G. A. Worth and H.-D. Meyer, The multiconfiguration time-dependent Hartree (MCTDH) method: A highly efficient algorithm for propagating wavepackets, Phys. Rep. 324 (2000), 1–105. 10.1016/S0370-1573(99)00047-2Suche in Google Scholar

[3] D. Braess and W. Hackbusch, Approximation of 1/x by exponential sums in [1,∞), IMA J. Numer. Anal. 25 (2005), no. 4, 685–697. 10.1093/imanum/dri015Suche in Google Scholar

[4] P. A. M. Dirac, Note on exchange phenomena in the Thomas atom, Math. Proc. Cambridge Philos. Soc. 26 (1930), 376–385. 10.1017/S0305004100016108Suche in Google Scholar

[5] L. Grasedyck, Existence and computation of low Kronecker-rank approximations for large linear systems of tensor product structure, Computing 72 (2004), no. 3–4, 247–265. 10.1007/s00607-003-0037-zSuche in Google Scholar

[6] L. Grasedyck, D. Kressner and C. Tobler, A literature survey of low-rank tensor approximation techniques, GAMM-Mitt. 36 (2013), no. 1, 53–78. 10.1002/gamm.201310004Suche in Google Scholar

[7] W. Hackbusch, Tensor Spaces and Numerical Tensor Calculus, Springer Ser. Comput. Math. 42, Springer, Berlin, 2012. 10.1007/978-3-642-28027-6Suche in Google Scholar

[8] W. Hackbusch, New estimates for the recursive low-rank truncation of block-structured matrices, Numer. Math. 132 (2016), no. 2, 303–328. 10.1007/s00211-015-0716-7Suche in Google Scholar

[9] E. Hairer, Solving differential equations on manifolds, Lecture notes, University of Geneva, Geneva, 2011. Suche in Google Scholar

[10] E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, Springer Ser. Comput. Math. 31, Springer, Berlin, 2002. 10.1007/978-3-662-05018-7Suche in Google Scholar

[11] E. Hairer, S. P. Nørsett and G. Wanner, Solving Ordinary Differential Equations. I: Nonstiff Problems, 2nd ed., Springer Ser. Comput. Math. 8, Springer, Berlin, 1993. Suche in Google Scholar

[12] T. Jahnke and W. Huisinga, A dynamical low-rank approach to the chemical master equation, Bull. Math. Biol. 70 (2008), no. 8, 2283–2302. 10.1007/s11538-008-9346-xSuche in Google Scholar PubMed

[13] E. Kieri, C. Lubich and H. Walach, Discretized dynamical low-rank approximation in the presence of small singular values, SIAM J. Numer. Anal. 54 (2016), no. 2, 1020–1038. 10.1137/15M1026791Suche in Google Scholar

[14] P. Kramer and M. Saraceno, Geometry of the Time-Dependent Variational Principle in Quantum Mechanics, Lecture Notes in Phys. 140, Springer, Berlin, 1981. 10.1007/3-540-10579-4Suche in Google Scholar

[15] D. Kressner and L. Periša, Recompression of Hadamard products of tensors in Tucker format, SIAM J. Sci. Comput. 39 (2017), no. 5, A1879–A1902. 10.1137/16M1093896Suche in Google Scholar

[16] D. Kressner, M. Steinlechner and B. Vandereycken, Preconditioned low-rank Riemannian optimization for linear systems with tensor product structure, SIAM J. Sci. Comput. 38 (2016), no. 4, A2018–A2044. 10.1137/15M1032909Suche in Google Scholar

[17] C. Lubich, From Quantum to Classical Molecular Dynamics: Reduced Models and Numerical Analysis, Zur. Lect. Adv. Math., European Mathematical Society (EMS), Zürich, 2008. 10.4171/067Suche in Google Scholar

[18] C. Lubich and I. V. Oseledets, A projector-splitting integrator for dynamical low-rank approximation, BIT 54 (2014), no. 1, 171–188. 10.1007/s10543-013-0454-0Suche in Google Scholar

[19] C. Lubich, I. V. Oseledets and B. Vandereycken, Time integration of tensor trains, SIAM J. Numer. Anal. 53 (2015), no. 2, 917–941. 10.1137/140976546Suche in Google Scholar

[20] 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), no. 2, 470–494. 10.1137/120885723Suche in Google Scholar

[21] 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), 3199–3213. 10.1063/1.463007Suche in Google Scholar

[22] H.-D. Meyer, F. Gatti and G. A. Worth, Multidimensional Quantum Dynamics: MCTDH Theory and Applications, Wiley, Weinheim, 2009. 10.1002/9783527627400Suche in Google Scholar

[23] H.-D. Meyer, U. Manthe and L. S. Cederbaum, The multi-configurational time-dependent Hartree approach, Chem. Phys. Lett. 165 (1990), 73–78. 10.1016/0009-2614(90)87014-ISuche in Google Scholar

[24] E. Musharbash, F. Nobile and T. Zhou, Error analysis of the dynamically orthogonal approximation of time dependent random PDEs, SIAM J. Sci. Comput. 37 (2015), no. 2, A776–A810. 10.1137/140967787Suche in Google Scholar

[25] A. Nonnenmacher and C. Lubich, Dynamical low-rank approximation: Applications and numerical experiments, Math. Comput. Simulation 79 (2008), no. 4, 1346–1357. 10.1016/j.matcom.2008.03.007Suche in Google Scholar

[26] I. V. Oseledets, Tensor-train decomposition, SIAM J. Sci. Comput. 33 (2011), no. 5, 2295–2317. 10.1137/090752286Suche in Google Scholar

[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), no. 5, 3744–3759. 10.1137/090748330Suche in Google Scholar

[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), no. 2, 1134–1162. 10.1137/110857520Suche in Google Scholar

[29] U. Schollwöck, The density-matrix renormalization group in the age of matrix product states, Ann. Physics 326 (2011), no. 1, 96–192. 10.1016/j.aop.2010.09.012Suche in Google Scholar

[30] A. Trombettoni and A. Smerzi, Discrete solitons and breathers with dilute Bose–Einstein condensate, Phys. Rev. Lett. 86 (2001), 2353–2356. 10.1103/PhysRevLett.86.2353Suche in Google Scholar PubMed

[31] A. Uschmajew and B. Vandereycken, The geometry of algorithms using hierarchical tensors, Linear Algebra Appl. 439 (2013), no. 1, 133–166. 10.1016/j.laa.2013.03.016Suche in Google Scholar

[32] J. H. Verner, Some Runge–Kutta formula pairs, SIAM J. Numer. Anal. 28 (1991), no. 2, 496–511. 10.1137/0728027Suche in Google Scholar

[33] F. Verstraete, J. J. García-Ripoll and J. I. Cirac, Matrix product density operators: Simulation of finite-temperature and dissipative systems, Phys. Rev. Lett. 93 (2004), Article ID 207204. 10.1103/PhysRevLett.93.207204Suche in Google Scholar PubMed

Received: 2017-09-29
Revised: 2018-04-12
Accepted: 2018-05-02
Published Online: 2018-07-21
Published in Print: 2019-01-01

© 2018 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. Tensor Numerical Methods: Actual Theory and Recent Applications
  3. A Low-Rank Inexact Newton–Krylov Method for Stochastic Eigenvalue Problems
  4. A Tensor Decomposition Algorithm for Large ODEs with Conservation Laws
  5. Non-intrusive Tensor Reconstruction for High-Dimensional Random PDEs
  6. Quasi-Optimal Rank-Structured Approximation to Multidimensional Parabolic Problems by Cayley Transform and Chebyshev Interpolation
  7. Projection Methods for Dynamical Low-Rank Approximation of High-Dimensional Problems
  8. Tensor Train Spectral Method for Learning of Hidden Markov Models (HMM)
  9. Tucker Tensor Analysis of Matérn Functions in Spatial Statistics
  10. Low-Rank Space-Time Decoupled Isogeometric Analysis for Parabolic Problems with Varying Coefficients
  11. Approximate Solution of Linear Systems with Laplace-like Operators via Cross Approximation in the Frequency Domain
  12. Rayleigh Quotient Methods for Estimating Common Roots of Noisy Univariate Polynomials
https://doi.org/10.1515/cmam-2018-0029
Schlagwörter für diesen Artikel
Tensor Train; Low-Rank Approximation; Tensor Differential Equations; Projection Methods

Artikel in diesem Heft

  1. Frontmatter
  2. Tensor Numerical Methods: Actual Theory and Recent Applications
  3. A Low-Rank Inexact Newton–Krylov Method for Stochastic Eigenvalue Problems
  4. A Tensor Decomposition Algorithm for Large ODEs with Conservation Laws
  5. Non-intrusive Tensor Reconstruction for High-Dimensional Random PDEs
  6. Quasi-Optimal Rank-Structured Approximation to Multidimensional Parabolic Problems by Cayley Transform and Chebyshev Interpolation
  7. Projection Methods for Dynamical Low-Rank Approximation of High-Dimensional Problems
  8. Tensor Train Spectral Method for Learning of Hidden Markov Models (HMM)
  9. Tucker Tensor Analysis of Matérn Functions in Spatial Statistics
  10. Low-Rank Space-Time Decoupled Isogeometric Analysis for Parabolic Problems with Varying Coefficients
  11. Approximate Solution of Linear Systems with Laplace-like Operators via Cross Approximation in the Frequency Domain
  12. Rayleigh Quotient Methods for Estimating Common Roots of Noisy Univariate Polynomials
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