  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.
