  In this paper, we present a predictor-corrector strategy for constructing
rank-adaptive dynamical low-rank approximations (DLRAs) of matrix-valued ODE
systems. The strategy is a compromise between (i) low-rank step-truncation
approaches that alternately evolve and compress solutions and (ii) strict DLRA
approaches that augment the low-rank manifold using subspaces generated locally
in time by the DLRA integrator. The strategy is based on an analysis of the
error between a forward temporal update into the ambient full-rank space, which
is typically computed in a step-truncation approach before re-compressing, and
the standard DLRA update, which is forced to live in a low-rank manifold. We
use this error, without requiring its full-rank representation, to correct the
DLRA solution. A key ingredient for maintaining a low-rank representation of
the error is a randomized singular value decomposition (SVD), which introduces
some degree of stochastic variability into the implementation. The strategy is
formulated and implemented in the context of discontinuous Galerkin spatial
discretizations of partial differential equations and applied to several
versions of DLRA methods found in the literature, as well as a new variant.
Numerical experiments comparing the predictor-corrector strategy to other
methods demonstrate robustness to overcome short-comings of step truncation or
strict DLRA approaches: the former may require more memory than is strictly
needed while the latter may miss transients solution features that cannot be
recovered. The effect of randomization, tolerances, and other implementation
parameters is also explored.
