An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation
Mathematics of Computation
Weighted essentially non-oscillatory schemes
Journal of Computational Physics
X-ray tomography in scattering media
SIAM Journal on Applied Mathematics
Least-Squares Finite-Element Solution of the Neutron Transport Equation in Diffusive Regimes
SIAM Journal on Numerical Analysis
The mathematics of computerized tomography
The mathematics of computerized tomography
Frequency Domain Optical Tomography Based on the Equation of Radiative Transfer
SIAM Journal on Scientific Computing
Sparse adaptive finite elements for radiative transfer
Journal of Computational Physics
Sparse tensor spherical harmonics approximation in radiative transfer
Journal of Computational Physics
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This paper presents a robust and accurate way to solve steady-state linear transport (radiative transfer) equations numerically. Our main objective is to address the inverse transport problem, in which the optical parameters of a domain of interest are reconstructed from measurements performed at the domain's boundary. This inverse problem has important applications in medical and geophysical imaging, and more generally in any field involving high frequency waves or particles propagating in scattering environments. Stable solutions of the inverse transport problem require that the singularities of the measurement operator, which maps the optical parameters to the available measurements, be captured with sufficient accuracy. This in turn requires that the free propagation of particles be calculated with care, which is a difficult problem on a Cartesian grid. A standard discrete ordinates method is used for the direction of propagation of the particles. Our methodology to address spatial discretization is based on rotating the computational domain so that each direction of propagation is always aligned with one of the grid axes. Rotations are performed in the Fourier domain to achieve spectral accuracy. The numerical dispersion of the propagating particles is therefore minimal. As a result, the ballistic and single scattering components of the transport solution are calculated robustly and accurately. Physical blurring effects, such as small angular diffusion, are also incorporated into the numerical tool. Forward and inverse calculations performed in a two-dimensional setting exemplify the capabilities of the method. Although the methodology might not be the fastest way to solve transport equations, its physical accuracy provides us with a numerical tool to assess what can and cannot be reconstructed in inverse transport theory.