The radiative transfer equation (RTE) occurs in a wide variety of applications. In this paper, we study discrete-ordinate discontinuous Galerkin methods for solving the RTE. The numerical methods are formed in two steps. In the first step, the discrete ordinate technique is applied to discretize the integral operator for the angular variable, resulting in a semi-discrete hyperbolic system. In the second step, the spatial discontinuous Galerkin method is applied to discretize the semi-discrete system. A stability and error analysis is performed on the numerical methods. Some numerical examples are included to demonstrate the convergence behavior of the methods.
The radiative transfer equation (RTE) arises in a variety of applications and is challenging to solve numerically due to its integro-differential form and high dimension. For highly forward-peaked media, it is even more difficult to solve RTE since accurate numerical solutions require a high resolution of the direction variable. For this reason, various approximations of RTE have been proposed in the literature. In this paper, we study a family of differential approximations of the RTE in three spatial variables. We explain the idea of constructing the differential approximations, and comment on the usefulness of the approximations.
Abstract. X-ray mammography is currently the most prevalent imaging modality for screening and diagnosis of breast cancers. However, its success is limited by the poor differentiation between healthy and diseased tissues in the mammogram. A potentially prominent imaging modality is based on the significant difference of x-ray scattering behaviors between tumor and normal tissues. Driven by major practical needs for better x-ray imaging, exploration into contrast mechanisms other than attenuation has been active for decades, e.g., in terms of scattering, which is also known as dark-field tomography. This paper provides a preliminary theoretical study of x-ray dark-field tomography (XDT) assuming the spectral x-ray detection technology. For XDT, the modified Leakeas-Larsen equation (MLLE) is an appropriate approximation of the radiative transfer equation (RTE) for a highly forward-peaked medium with small but sufficient amounts of large-angle scattering. Properties of the MLLE are studied, such as existence of a unique solution and positivity of the solution. MLLE and its discrete analogues can be solved naturally with an iteration procedure, and convergence of the iteration procedure is shown. XDT, as an inverse parameter problem with MLLE as the forward model, is then studied. Numerical discretization schemes of MLLE and the associated XDT are introduced. Simulation results are reported on several numerical examples for MLLE and for XDT. The paper concludes with some remarks on research topics for further study of XDT. 1. Introduction. X-ray mammography is currently the most prevalent imaging modality for screening and diagnosis of breast cancers. The use of mammography results in a 25-30% decreased mortality rate in screened women [25]. However, a multi-institutional trial funded by the American College of Radiology Imaging Network (ACRIN) suggested that about 30% of cancers were not detected by screening mammography (false negatives), and 70-90% of biopsies performed based on suspicious mammograms were negative (false positives) [29]. The key factor that limits mammography success rate is the poor differentiation between healthy and diseased tissues in the mammogram. Although x-ray computed tomography (CT) of the breast can potentially improve diagnostic accuracy over mammography [14,15], the stateof-the-art breast CT scanner is still based on the attenuation mechanism. As a result, the use of breast CT requires an intravenous contrast medium and a high radiation
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