Volume 57, Number 2, March-April 2023
|Page(s)||865 - 891|
|Published online||30 March 2023|
A robust collision source method for rank adaptive dynamical low-rank approximation in radiation therapy
Karlsruhe Institute of Technology, Computational Science and Mathematical Methods, Karlsruhe, Germany
2 Karlsruhe Institute of Technology, Steinbuch Centre for Computing, Karlsruhe, Germany
3 German Cancer Research Center - DKFZ, Department of Medical Physics in Radiation Oncology, Heidelberg, Germany
4 HIDSS4Health - Helmholtz Information and Data Science School for Health, Karlsruhe/Heidelberg, Germany
* Corresponding author: firstname.lastname@example.org
Accepted: 4 November 2022
Deterministic models for radiation transport describe the density of radiation particles moving through a background material. In radiation therapy applications, the phase space of this density is composed of energy, spatial position and direction of flight. The resulting six-dimensional phase space prohibits fine numerical discretizations, which are essential for the construction of accurate and reliable treatment plans. In this work, we tackle the high dimensional phase space through a dynamical low-rank approximation of the particle density. Dynamical low-rank approximation (DLRA) evolves the solution on a low-rank manifold in time. Interpreting the energy variable as a pseudo-time lets us employ the DLRA framework to represent the solution of the radiation transport equation on a low-rank manifold for every energy. Stiff scattering terms are treated through an efficient implicit energy discretization and a rank adaptive integrator is chosen to dynamically adapt the rank in energy. To facilitate the use of boundary conditions and reduce the overall rank, the radiation transport equation is split into collided and uncollided particles through a collision source method. Uncollided particles are described by a directed quadrature set guaranteeing low computational costs, whereas collided particles are represented by a low-rank solution. It can be shown that the presented method is L2-stable under a time step restriction which does not depend on stiff scattering terms. Moreover, the implicit treatment of scattering does not require numerical inversions of matrices. Numerical results for radiation therapy configurations as well as the line source benchmark underline the efficiency of the proposed method.
Mathematics Subject Classification: 65F55 / 65M12 / 65M08 / 65M70
Key words: Dynamical low-rank approximation / radiation therapy / kinetic equations / rank adaptivity / model order reduction
© The authors. Published by EDP Sciences, SMAI 2023
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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