Discretized Dynamical Low-Rank Approximation in the Presence of Small Singular Values

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 -perturbation of a low-rank matrix or tensor train, the error of the projector-splitting integrator is favorably bounded in terms of 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.