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Computational Methods in Applied Mathematics

Editor-in-Chief: Carstensen, Carsten

Managing Editor: Matus, Piotr


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Volume 19, Issue 1

Issues

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

Emil Kieri / Bart Vandereycken
Published Online: 2018-07-21 | DOI: https://doi.org/10.1515/cmam-2018-0029

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

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About the article

Received: 2017-09-29

Revised: 2018-04-12

Accepted: 2018-05-02

Published Online: 2018-07-21

Published in Print: 2019-01-01


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


Citation Information: Computational Methods in Applied Mathematics, Volume 19, Issue 1, Pages 73–92, ISSN (Online) 1609-9389, ISSN (Print) 1609-4840, DOI: https://doi.org/10.1515/cmam-2018-0029.

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