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

by E. Kieri, B. Vandereycken

#### Abstract:

We consider dynamical low-rank approximation on the manifold of xed-rank tensor trains (also called matrix product states), and analyse projection methods for the time integration of such problems. First, we prove error estimates for the explicit Euler method, amended with quasi-optimal projections to the manifold, under suitable approximability assumptions. Then, we discuss the possibilities and di culties with higher order explicit and implicit projected Runge--Kutta 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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#### Reference:

E. Kieri, B. Vandereycken, "Projection Methods for Dynamical Low-Rank Approximation of High-Dimensional Problems", In Comput. Meth. Appl. Math., vol. 19, no. 1, pp. 73-92, 2018.

#### Bibtex Entry:

@article{Kieri_V:2018, Abstract = {We consider dynamical low-rank approximation on the manifold of xed-rank tensor trains (also called matrix product states), and analyse projection methods for the time integration of such problems. First, we prove error estimates for the explicit Euler method, amended with quasi-optimal projections to the manifold, under suitable approximability assumptions. Then, we discuss the possibilities and di culties with higher order explicit and implicit projected Runge--Kutta methods. In particular, we discuss ways for limiting rank growth in the increments, and robustness with respect to small singular values.}, Author = {Kieri, E. and Vandereycken, B.}, Journal = {Comput. Meth. Appl. Math.}, Number = {1}, Pages = {73--92}, Title = {Projection Methods for Dynamical Low-Rank Approximation of High-Dimensional Problems}, Volume = {19}, Year = {2018}, Doi = {10.1515/cmam-2018-0029}, Pdf = {http://www.unige.ch/math/vandereycken/papers/published_Kieri_V_2018.pdf} }