It is of great interest to solve the inverse problem of stationary radiative transport equation (RTE) in optical tomography. The standard way is to formulate the inverse problem into an optimization problem, but the bottleneck is that one has to solve the forward problem repeatedly, which is time-consuming. Due to the optical property of biological tissue, in real applications, optical thin and thick regions coexist and are adjacent to each other, and the geometry can be complex. To use coarse meshes and save the computational cost, the forward solver has to be asymptotic preserving across the interface (APAL). In this paper, we propose an offline/online solver for RTE. The cost at the offline stage is comparable to classical methods, while the cost at the online stage is much lower. Two cases are considered. One is to solve the RTE with fixed scattering and absorption cross sections while the boundary conditions vary; the other is when cross sections vary in a small domain and the boundary conditions change many times. The solver can be decomposed into offline/online stages in these two cases. One only needs to calculate the offline stage once and update the online stage when the parameters vary. Our proposed solver is much cheaper when one needs to solve RTE with multiple right-hand sides or when the cross sections vary in a small domain, thus can accelerate the speed of solving inverse RTE problems. We illustrate the online/offline decomposition based on the Tailored Finite Point Method (TFPM), which is APAL on general quadrilateral meshes.

翻译：在光学断层成像中，求解稳态辐射输运方程（RTE）的反问题具有重要研究价值。标准方法是将反问题转化为优化问题，但瓶颈在于需要反复求解正演问题，导致计算耗时。由于生物组织的光学特性，实际应用中光学薄区和厚区常共存且相互毗邻，几何结构可能复杂。为使用粗网格并节省计算成本，正演求解器必须具有界面渐近保持特性（APAL）。本文提出一种适用于RTE的离/在线求解器：离线阶段成本与传统方法相当，而在线阶段成本显著降低。考虑两种情形：一是固定散射和吸收截面但边界条件变化时求解RTE；二是截面在小区域内变化且边界条件多次改变。在这两种情形下，求解器均可分解为离/在线阶段，仅需计算一次离线阶段，当参数变化时更新在线阶段即可。当需要求解多个右端项的RTE或截面在小区域变化时，本文所提求解器成本显著降低，从而可加速RTE反问题的求解速度。基于在通用四边形网格上具有APAL特性的定制有限点方法（TFPM），我们阐释了这种在线/离线分解技术。