| Abstract: |
| In this talk, I present a predictor-corrector strategy for constructing rank-adaptive dynamical low-rank approximations of matrix-valued differential equation systems. Dynamical low-rank approximation (DLRA) is a nonlinear model reduction technique that evolves dynamical systems on a low-rank manifold, and has recently become popular in the approximation of linear and non-linear partial differential equations from kinetic theory. The strategy presented 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. This error is used, 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, 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 shortcomings 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. |
|