Dynamical low-rank integrators

Within projects A7 and B9, dynamical low-rank integrators (DLRIs) play an essential role. Several new fixed-rank and rank-adaptive DLRIs have been proposed, implemented, and tested. The following list provides an overview of all publications concerned with DLRIs and also collects the links to the respective codes which are used to create the data.

Requirements

Details about the specific requirements can be found in the Readme and Instruction files of the respective code.

TITUS

TITUS is a three-dimensional GPU accelerated radiation transport code for electron and proton therapy. In contrast to conventional deterministic codes, it uses dynamical low-rank approximation (DLRA) to enable an efficient representation of the radiation transport solution. TITUS incorporates several new DLRA functionalities that have been developed in the CRC. These include the rank-adaptive "unconventional" Basis update & Galerkin integrator, stable high-order spatial discretizations for the DLRA evolution equations and collided-uncollided splitting strategies. Further information on how to use the software can be found on the github page as well as the original article A robust collision source method for rank adaptive dynamical low-rank approximation in radiation therapy.

Software to reproduce numerical results in publications related to dynamical low rank approximations

• https://github.com/JonasKu/publication-A-robust-collision-source-method-for-rank-adaptive-dynamical-low-rankapproximation.git
  Info: This repository contains the parts of the TITUS software needed to reproduce the results in A robust collision source method for rank adaptive dynamical low-rank approximation in radiation therapy.
• https://github.com/JonasKu/Publication-A-low-rank-power-iteration-scheme-for-neutron-transport-criticalityproblems.git
  Info: This code implements A low-rank power iteration scheme for neutron transport criticality problems in nuclear systems.
• https://github.com/JonasKu/publication-A-rank-adaptive-robust-integrator-for-dynamical-low-rank-approximation.git
  Info: This repository contains the source code for A rank-adaptive robust integrator for dynamical low-rank approximation.
  Numerical examples range from 1D and 2D problems in radiation transport to Schrödinger equations as well as hyperbolic uncertainty quantification.
• https://github.com/JonasKu/Testcases-for-Technical-Report---Dynamical-low-rank-approximation-for-Marshak-waves.git
  Info: This code is a discontinuous Galerkin implementation for Marshak wave problems with dynamical low-rank approximation. The code includes the projector-splitting integrator as well as the rank-adaptive unconventional integrator, see Dynamical low-rank approximation for Marshak waves.
• https://github.com/JonasKu/publication-On-the-stability-of-robust-dynamical-low-rank-approximations.git
  Info: This code can be used to reproduce all results from On the stability of robust dynamical low-rank approximations for hyperbolic problems (to be published in SISC). It provides a collection of DLRA methods as well as their corresponding discretization strategies and record the discrete $L^2$ norm to indicate which discretizations are stable and which are not.
• https://github.com/JonasKu/publication-Dynamical-low-rank-approximation-for-Burgers-equation-with-uncertainty.git
  Info: This code framework provides an implementation of a Dynamical low-rank approximation for Burgers' equation with uncertainty. The code uses dynamical low-rank approximation to reduce computational costs and memory footprint in uncertainty quantification. Implementations include the projector-splitting integrator and the unconventional integrator.
• Numerical experiments for "On dynamical low-rank integrators for matrix differential equations"
  Info: These codes can be used to reproduce all results from the doctoral thesis On dynamical low-rank integrators for matrix differential equations by Stefan Schrammer.
• Numerical experiments to "Rank-adaptive dynamical low-rank integrators for first-order and second-order matrix differential equations"
  Info: These are the codes which were used to create the data and figures in the CRC Preprint Rank-adaptive dynamical low-rank integrators for first-order and second-order matrix differential equations.
• Numerical experiments for "Dynamical low-rank integrators for second-order matrix differential equations"
  Info: The link provides an implementation of dynamcial low-rank integrators for second-order matrix differential equations. Moreover, two numerical experiments (homogeneous wave equation, wave equation with cubic nonlinearity) illustrate the behavior of the respective schemes.