Robust time integration for dynamical low-rank approximation

The talk begins with a brief review of dynamical low-rank approximation for matrix (and tensor) differential equations, emphasizing the difficulties created by the ubiquitous  appearance of small singular values. These difficulties are overcome by the projector-splitting integrator, which splits the projection onto the tangent space of the low-rank manifold into an alternating sum of subprojections. It has a surprising exactness property and is robust to small singular values. A very recent robust integrator is the ``unconventional'' low-rank integrator, which uses a basis-update and Galerkin approach. It shares the robust error bounds with the projector-splitting integrator but is more parallel and avoids the backward substep that appears problematic for strongly dissipative problems. Augmenting the Galerkin step to the larger subspace generated by both the old and new bases and using a tolerance-controlled rank truncation after the step yields a novel rank-adaptive integrator with remarkable (near-)conservation properties. Numerical experiments illustrate the theoretical results.
While the talk will be restricted to dynamical low-rank approximation for matrix differential equations in order to highlight basic construction principles, the methods discussed here can all be nontrivially extended to robust time integration methods for general tree tensor network approximations.

This talk is based on joint work with Ivan Oseledets and with Emil Kieri and Hanna Walach (projector-splitting integrator), with Gianluca Ceruti (unconventional integrator, rank-adaptive integrator) and Jonas Kusch (rank-adaptive integrator). Recent work on time integration of general tree tensor networks is jointly with Hanna Walach, Gianluca Ceruti and Dominik Sulz.