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

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 exp...

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Bibliographic Details
Published in:Journal of computational methods in applied mathematics Vol. 19; no. 1; pp. 73 - 92
Main Authors: Kieri, Emil, Vandereycken, Bart
Format: Journal Article
Language:English
Published: Minsk De Gruyter 01.01.2019
Walter de Gruyter GmbH
Subjects:
58J35
65L05
65L06
65L70
Approximation
Low-Rank Approximation
Manifolds (mathematics)
Mathematical analysis
Projection Methods
Tensor Differential Equations
Tensor Train
Tensors
Time integration
ISSN:1609-4840, 1609-9389
Online Access:Get full text
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Summary: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.
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ISSN:1609-4840
1609-9389
DOI:10.1515/cmam-2018-0029