The dynamical low-rank approximation (DLRA) is a technique for solving large-scale matrix differential equations at a fraction of the original costs.

Our first contribution is a parallel-in-time method called low-rank Parareal in which we develop and study the parallelization of DLRAs. We prove the linear and superlinear convergence of the method. To our knowledge, it is the first parallel-in-time integrator for the DLRA.

Our second contribution introduces projected exponential methods for stiff problems with a Sylvester-like structure. We focus on two cases for which we prove robust first and second order convergence with respect to the timestep. All our results are verified numerically.

Finally, we explore novel dynamical randomized methods for DLRAs. We propose two methods called dynamical randomized SVD and dynamical generalized Nyström. Our experiments show a good performance of the methods and seem to indicate that the methods preserve critical physical quantities such as the energy and geometry of the solution.