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.
翻译:辐射传输的确定性模型描述通过背景材料移动的辐射粒子密度。 在辐射治疗应用中, 这种密度的阶段空间由能量、空间位置和飞行方向组成。 由此产生的六维相位空间禁止细数字分解, 这对于构建准确和可靠的治疗计划至关重要。 在这项工作中, 我们通过粒子密度动态低端近似来应对高维阶段空间。 动态低端近距离( DLRA) 将解决方案在低端的时序中演变为低层的粒子。 在将能量变量解释为假时, 允许我们使用 DLRA 框架来代表每个能源低层的低层传输线方程式的解决方案。 通过高效的隐含性能源分解和等级的适应性分解来处理高维空间。 为了便于使用边界条件和降低总体水平, 动态低层近距离近距离近距离近( DLLLL) 通过碰撞源方法将解解成混凝质的粒子。 低层粒子的分解性粒子由定向的平流法来描述, 保证低层分解的分流法是低层的分解法, 。 以低序法 以低序法 代表着的分解的分解法 。 。, 以低序法可以显示的分解的分解的分解的分解的分解法 。