A Predictor-Corrector Strategy for Adaptivity in Dynamical Low-Rank Approximations

Dr. Stefan Schnake

Abstract: In this talk, I present a predictor-corrector strategy for constructing rank-adaptive dynamical low-rank approximations (DLRAs) of matrix-valued ordinary differential equation 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. 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.

Speaker’s Bio: Stefan Schnake is a postdoctoral research associate in the Multiscale Methods and Dynamics Group. He received his Ph.D. in 2017 from the University of Tennessee and joined the Oak Ridge National Laboratory in 2020 after a postdoctoral appointment at the University of Oklahoma. His research interests include low-rank and sparse methods for kinetic problems and discontinuous Galerkin methods for non-standard elliptic partial differential equations.

Last Updated: September 13, 2022 - 8:46 am