Abstract. Low-rank approximations to large time-dependent matrices and tensors are the subject of this paper. These matrices and tensors either are given explicitly or are the unknown solutions of matrix and tensor differential equations. Based on splitting the orthogonal projection onto the tangent space of the low-rank manifold, novel time integrators for obtaining approximations by low-rank matrices and low-rank tensor trains were recently proposed. By standard theory, the Lie-Trotter and Strang projector-splitting methods are first and second order accurate, respectively, but the usual error bounds break down when the low-rank approximation has small singular values. This happens when the singular values of the solution decay without a distinct gap or when the effective rank of the solution is overestimated. On the other hand, the integrators are exact when given time-dependent matrices or tensors are already of the prescribed rank. We provide an error analysis which unifies these properties. We show that in cases where the exact solution is an ε-perturbation of a low-rank matrix or tensor train, the error of the projector-splitting integrator is favorably bounded in terms of ε and the stepsize, independently of the smallness of the singular values. Such a result does not hold for any standard integrator. Numerical experiments illustrate the theory.Key words. tensor train, low-rank approximation, tensor differential equations, splitting integrator AMS subject classifications. 15A18, 65L05, 65L70 DOI. 10.1137/15M10267911. Introduction. Low-rank approximations to matrices and tensors are a basic tool in data and model reduction; see, e.g., [4,5]. In this paper we are concerned with the low-rank approximation of time-dependent matrices and tensors, which either are given explicitly by their increments or are the unknown solutions of differential equations. In [11] such time-dependent problems and their numerical treatment were first studied for matrices. Differential equations for the factors of a low-rank factorization similar to the singular value decomposition were derived and their approximation properties were studied. Extensions to time-dependent tensors in various tensor formats were given in [2,12,18,19]; see also [15] for a review of dynamical low-rank approximation.The approach yields differential equations on low-rank matrix and tensor manifolds, which need to be solved numerically. Recently, very efficient integrators based on splitting the projection onto the tangent space of the low-rank manifold have been proposed and studied for matrices and for tensors in the tensor-train format in [16] and [17], respectively. The objective of the present paper is to show that these projector-splitting integrators are insensitive to the presence of small singular values

Dynamical low-rank approximation in the Tucker tensor format of given large timedependent 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 favorable 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.

Dynamical low-rank approximation by tree tensor networks is studied for the datasparse approximation to large time-dependent data tensors and unknown solutions of tensor differential equations. A time integration method for tree tensor networks of prescribed tree rank is presented and analyzed. It extends the known projector-splitting integrators for dynamical low-rank approximation by matrices and Tucker tensors and is shown to inherit their favorable properties. The integrator is based on recursively applying the Tucker tensor integrator. In every time step, the integrator climbs up and down the tree: it uses a recursion that passes from the root to the leaves of the tree for the construction of initial value problems on subtree tensor networks using appropriate restrictions and prolongations, and another recursion that passes from the leaves to the root for the update of the factors in the tree tensor network. The integrator reproduces given time-dependent tree tensor networks of the specified tree rank exactly and is robust to the typical presence of small singular values in matricizations of the connection tensors, in contrast to standard integrators applied to the differential equations for the factors in the dynamical low-rank approximation by tree tensor networks.

We propose a numerical integrator for determining low-rank approximations to solutions of large-scale matrix differential equations. The considered differential equations are semilinear and stiff. Our method consists of first splitting the differential equation into a stiff and a non-stiff part, respectively, and then following a dynamical low-rank approach. We conduct an error analysis of the proposed procedure, which is independent of the stiffness and robust with respect to possibly small singular values in the approximation matrix. Following the proposed method, we show how to obtain low-rank approximations for differential Lyapunov and for differential Riccati equations. Our theory is illustrated by numerical experiments.

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