  This work introduces a parallel and rank-adaptive matrix integrator for
dynamical low-rank approximation. The method is related to the previously
proposed rank-adaptive basis update & Galerkin (BUG) integrator but differs
significantly in that all arising differential equations, both for the basis
and the Galerkin coefficients, are solved in parallel. Moreover, this approach
eliminates the need for a potentially costly coefficient update with augmented
basis matrices. The integrator also incorporates a new step rejection strategy
that enhances the robustness of both the parallel integrator and the BUG
integrator. By construction, the parallel integrator inherits the robust error
bound of the BUG and projector-splitting integrators. Comparisons of the
parallel and BUG integrators are presented by a series of numerical experiments
which demonstrate the efficiency of the proposed method, for problems from
radiative transfer and radiation therapy.

Numerical Analysis;Numerical Analysis