Rank-adaptive dynamical low-rank integrators for first-order and second-order matrix differential equations

Abstract

Dynamical low-rank integrators for matrix differential equations recently attracted a lot of attention and have proven to be very efficient in various applications. In this paper, we propose a novel strategy for choosing the rank of the projector-splitting integrator of Lubich and Oseledets adaptively. It is based on a combination of error estimators for the local time-discretization error and for the low-rank error with the aim to balance both. This ensures that the convergence of the underlying time integrator is preserved. The adaptive algorithm works for projector-splitting integrator methods for first-order matrix differential equations and also for dynamical low-rank integrators for second-order equations, which use the projector-splitting integrator method in its substeps. Numerical experiments illustrate the performance of the new integrators.

A rank-adaptive robust integrator for dynamical low-rank approximation

“…Although second order is widely observed for the Strang projector splitting (see, e.g., [8,31]), the known proof of robust convergence only yields order 1 [35]. The different variants of the BUG integrators also have robust first-order error bounds [11,9,10].…”

“…Dynamical low-rank approximation of time-dependent matrices [37] has proven to be an efficient model order reduction technique for applications from widely varying fields including plasma physics [24,26,20,8,22,14,27,15,23,56], radiation transport [49,16,47,41,48,57,21,4], radiation therapy [40], chemical kinetics [32,50,25], wave propagation [31,58], kinetic shallow water models [38], uncertainty quantification [51,3,28,43,44,46,39,33,17,2], and machine learning [54,59,52,53]. These problems can be written as a prohibitively large matrix differential equation for A(t) ∈ R m×n , .…”

A robust and conservative dynamical low-rank algorithm

“…These methods originate from early work in quantum mechanics [46,41] and were later considered in a mathematical framework for for ordinary differential equations (see, e.g., [35,47]). In the latter context, many advances such as robust integrators [42,5], rank adaptive methods [6,7,33,32], and generalization to various tensor formats [43,44,45,14,4,7] have been made. While such methods can be applied in a rather generic way to ordinary or partial differential equation, an efficient algorithm is only obtained if a suitable decomposition of variables is chosen that allows us to run the simulation with a small to moderate rank.…”

On the Stability of Robust Dynamical Low-Rank Approximations for Hyperbolic Problems

“…Though dynamical low-rank approximation (DLRA) [33] offers a significant reduction of computational costs and memory consumption when solving tensor differential equations [26,10,54], the use of DLRA to solve matrix differential equations has sparked immense interest in several communities. Research fields in which DLRA for matrix differential equations has a considerable impact include plasma physics [21,23,16,5,18,12,24,13,20,55], radiation transport [47,14,45,39,46,56,17,3,38], chemical kinetics [29,48,22], wave propagation [28,57], uncertainty quantification [49,2,25,41,42,43,36,30,15,1], and machine learning [52,58,50,51]. These application fields commonly require memory-intensive and computational costly numerical simulations due to the solution's prohibitively large phase space.…”