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

Abstract

The dynamical low-rank approximation (DLRA) is used to treat high-dimensional problems that arise in such diverse fields as kinetic transport and uncertainty quantification. Even though it is well known that certain spatial and temporal discretizations when combined with the DLRA approach can result in numerical instability, this phenomenon is poorly understood. In this paper we perform a L 2 stability analysis for the corresponding nonlinear equations of motion. This reveals the source of the instability for the projector splitting integrator when first discretizing the equations and then applying the DLRA. Based on this we propose a projector splitting integrator, based on applying DLRA to the continuous system before performing the discretization, that recovers the classic CFL condition. We also show that the unconventional integrator has more favorable stability properties and explain why the projector splitting integrator performs better when approximating higher moments, while the unconventional integrator is generally superior for first order moments. Furthermore, an efficient and stable dynamical low-rank update for the scattering term in kinetic transport is proposed. Numerical experiments for kinetic transport and uncertainty quantification, which confirm the results of the stability analysis, are presented.

A robust and conservative dynamical low-rank algorithm

“…First, it makes the distinction between the error due to the low-rank approximation and the error due to the spatial and temporal discretization clear. Second, since we start from a set of partial differential equations for the low-rank factors by providing an appropriate space and temporal discretization we can avoid the stability issues that in some cases arises if the numerical discretization is performed first [38]. Third, the numerical discretization can be tailored to the partial differential equations obtained for the low-rank factors.…”

“…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. For kinetic problems primarily a decomposition between spatial and velocity variables has been performed (see [16,22,23,49,48,12,8,38]; some work on tensor decomposition also exists [36,19,1,28]). It turns out that for kinetic problems this has a number of advantages.…”

Macro-micro decomposition for consistent and conservative model order reduction of hyperbolic shallow water moment equations: A study using POD-Galerkin and dynamical low rank approximation

“…Stable time integrators for the resulting DLRA system, which are robust irrespective of the curvature of the low-rank manifold [25], are the matrix projector-splitting integrator [40] as well as the "unconventional" basis update & Galerkin step (BUG) integrator of [7]. Here, we use the BUG integrator which only evolves the solution forward in time, thereby facilitating the construction of stable spatial discretizations [35]. Moreover, the BUG integrator enables a straightforward basis augmentation step [6] which simplifies the construction of rank adaptive methods [6,36,20] and allows for conservation properties [6,14].…”

Adaptation of the urban climate model PALM-4U for Wuerzburg, Germany

“…DLRA's core idea is to represent and evolve the solution on the low-rank manifold of rank r functions. Past work in the area of radiative transfer has focused on asymptotic-preserving schemes [9,8], mass conservation [29], stable discretizations [18], imposing boundary conditions [19,15] and implicit time discretizations [30]. A discontinuous Galerkin discretization of the DLRA evolution equations for thermal radiative transfer has been proposed in [4].…”

“…Three integrators which move on the low-rank manifold while not being restricted by its curvature are the projector-splitting (PS) integrator [22], the basis update & Galerkin (BUG) integrator [7], and the parallel integrator [6]. Since the PS integrator evolves one of the required subflows backward in time, the BUG and parallel integrator are preferable for diffusive problems while facilitating the construction of stable numerical discretization for hyperbolic problems [18]. Moreover, the BUG integrator allows for a basis augmentation step [5] which can be used to construct conservative schemes for the Schrödinger equation [5] and the Vlasov-Poisson equations [11].…”