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

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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We consider dynamical low-rank approximations to parabolic problems on higher-order tensor manifolds in Hilbert spaces. In addition to existence of solutions and their stability with respect to perturbations to the problem data, we show convergence of spatial discretizations. Our framework accommodates various standard low-rank tensor formats for multivariate functions, including tensor train and hierarchical tensors.