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.
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
and thereby
Now take any
We choose
Since
and by (A.1) and
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.
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