Linear kinetic transport equations play a critical role in optical tomography, radiative transfer and neutron transport. The fundamental difficulty hampering their efficient and accurate numerical resolution lies in the high dimensionality of the physical and velocity/angular variables and the fact that the problem is multiscale in nature. Leveraging the existence of a hidden low-rank structure hinted by the diffusive limit, in this work, we design and test the angular-space reduced order model for the linear radiative transfer equation, the first such effort based on the celebrated reduced basis method (RBM). Our method is built upon a high-fidelity solver employing the discrete ordinates method in the angular space, an asymptotic preserving upwind discontinuous Galerkin method for the physical space, and an efficient synthetic accelerated source iteration for the resulting linear system. Addressing the challenge of the parameter values (or angular directions) being coupled through an integration operator, the first novel ingredient of our method is an iterative procedure where the macroscopic density is constructed from the RBM snapshots, treated explicitly and allowing a transport sweep, and then updated afterwards. A greedy algorithm can then proceed to adaptively select the representative samples in the angular space and form a surrogate solution space. The second novelty is a least-squares density reconstruction strategy, at each of the relevant physical locations, enabling the robust and accurate integration over an arbitrarily unstructured set of angular samples toward the macroscopic density. Numerical experiments indicate that our method is effective for computational cost reduction in a variety of regimes.

翻译：在光学透视、辐射传输和中子传输中,线性动能传输方程式发挥着关键作用。阻碍其高效和准确数字解析的根本困难在于物理和速度/角变量的高度维度,而且问题在于其性质是多尺度的。利用由分流限制暗示的隐藏的低级别结构的存在,在这项工作中,我们设计和测试线性辐射传输方程式的角-空间降序模型,这是以已知的降低基数法(RBM)为基础的首个此类工作。我们的方法建立在高纤维解析解析器上,在角空间使用离散的内径坐标变量解解度解析法,在物理空间上和速度/角变异性变异性变异性变异性变异性变异性变异性变异性变异性变异性变异性变异性方法中,在随后,通过整合操作一个不精确和直径直径变异性变异性变异性变异性变异性变异性变异性变异性变异性变异性变异性变性变异性变异性变异性变异性变异性变性变异性变异性变性变性变异性变异性变性变性变异性变异性变异性变性变性变性变性变性方法,在一次性变异性变异性变性变性变性变性变性变性变性变异性变性变性变性变性变性变性变性变性变性变性变性变性变性变性变异性变性变性变性变性变性变异性变性变性变异性变性变性变异性变异性变异性变性变性变性变性变性变性变性变性变性变性变性变性变性变异性变性变性变性变性变性变性变性变性变性变性变性变性变性变性变性变性变性变性变性变性变性变性变性变性变性变性变性变性变性变性变性变性变性变性变性变性变性变性变性变性变性变性变性变性变性变性变性变性