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Sparse tensor phase space Galerkin approximation for radiative transport

by K. Grella

(Report number 2014-03)

Abstract
We develop, analyze, and test a sparse tensor product phase space Galerkin discretization framework for the stationary monochromatic radiative transfer problem with scattering. The mathematical model describes the transport of radiation on a phase space of the Cartesian product of a typically three-dimensional physical domain and two-dimensional angular domain. Known solution methods such as the discrete ordinates method and a spherical harmonics method are derived from the presented Galerkin framework. We construct sparse versions of these well-established methods from the framework and prove that these sparse tensor discretizations break the ``curse of dimensionality'': essentially (up to logarithmic factors in the total number of degrees of freedom) the solution complexity increases only as in a problem posed in the physical domain alone, while asymptotic convergence orders in terms of the discretization parameters remain essentially equal to those of a full tensor phase space Galerkin discretization. Algorithmically we compute the sparse tensor approximations by the combination technique. In numerical experiments on \(2+1\) and \(3+2\) dimensional phase spaces we demonstrate that the advantages of sparse tensorization can be leveraged in applications.

Keywords: radiative transfer; sparse grids; discrete ordinates method; spherical harmonics method; combination technique

@Techreport{G14_553,
  author = {K. Grella},
  title = {Sparse tensor phase space Galerkin approximation for radiative transport},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2014-03},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2014/2014-03.pdf },
  year = {2014}
}

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