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@ARTICLE{Kusch:275783,
      author       = {J. Kusch and P. Stammer$^*$},
      title        = {{A} robust collision source method for rank adaptive
                      dynamical low-rank approximation in radiation therapy},
      journal      = {Mathematical modelling and numerical analysis},
      volume       = {57},
      number       = {2},
      issn         = {2822-7840},
      address      = {Les Ulis},
      publisher    = {EDP Sciences},
      reportid     = {DKFZ-2023-00877},
      pages        = {865 - 891},
      year         = {2023},
      note         = {#LA:E040#},
      abstract     = {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 L-2-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.},
      cin          = {E040},
      ddc          = {510},
      cid          = {I:(DE-He78)E040-20160331},
      pnm          = {315 - Bildgebung und Radioonkologie (POF4-315)},
      pid          = {G:(DE-HGF)POF4-315},
      typ          = {PUB:(DE-HGF)16},
      doi          = {10.1051/m2an/2022090},
      url          = {https://inrepo02.dkfz.de/record/275783},
}