Bart Vandereycken

Time integration of rank-constrained Tucker tensors

by , ,

Abstract:

Dynamical low-rank approximation in the Tucker format of given large time-dependent tensors and of tensor differential equations is the subject of this paper. In particular, a discrete time integration method for rank- constrained Tucker tensors is presented and analyzed. It extends the known projector-splitting integrator for dynamical low-rank approximation of matrices to Tucker tensors and is shown to inherit the same favor- able properties. The integrator is based on iteratively applying the matrix projector-splitting integrator to tensor unfoldings but with inexact solution in a substep. It has the property that it reconstructs time-dependent Tucker tensors of the given rank exactly. The integrator is also shown to be robust to the presence of small singular values in the tensor unfoldings. Numerical examples with problems from quantum dynamics and tensor optimization methods illustrate our theoretical results.

Reference:

Ch. Lubich, H. Walach, B. Vandereycken, "Time integration of rank-constrained Tucker tensors", In SIAM J. Numer. Anal., vol. 56, no. 3, pp. 1273-1290, 2018.

Bibtex Entry:

@article{Lubich_WV:2018,
	Abstract = {Dynamical low-rank approximation in the Tucker format of given large time-dependent tensors and of tensor differential equations is the subject of this paper. In particular, a discrete time integration method for rank- constrained Tucker tensors is presented and analyzed. It extends the known projector-splitting integrator for dynamical low-rank approximation of matrices to Tucker tensors and is shown to inherit the same favor- able properties. The integrator is based on iteratively applying the matrix projector-splitting integrator to tensor unfoldings but with inexact solution in a substep. It has the property that it reconstructs time-dependent Tucker tensors of the given rank exactly. The integrator is also shown to be robust to the presence of small singular values in the tensor unfoldings. Numerical examples with problems from quantum dynamics and tensor optimization methods illustrate our theoretical results.},
	Author = {Lubich, Ch. and Walach, H. and Vandereycken, B.},
    Journal = {SIAM J. Numer. Anal.},
    Number = {3},
    Pages = {1273--1290},
    Title = {Time integration of rank-constrained {T}ucker tensors},
    Volume = {56},
    Year = {2018},    
    Doi = {10.1137/17M1146889},
    Pdf = {http://www.unige.ch/math/vandereycken/papers/published_Lubich_VW_2018.pdf}
}