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It is of great interest to solve the inverse problem of stationary radiativetransport equation (RTE) in optical tomography. The standard way is toformulate the inverse problem into an optimization problem, but the bottleneckis that one has to solve the forward problem repeatedly, which istime-consuming. Due to the optical property of biological tissue, in realapplications, optical thin and thick regions coexist and are adjacent to eachother, and the geometry can be complex. To use coarse meshes and save thecomputational cost, the forward solver has to be asymptotic preserving acrossthe interface (APAL).  In this paper, we propose an offline/online solver for RTE. The cost at theoffline stage is comparable to classical methods, while the cost at the onlinestage is much lower. Two cases are considered. One is to solve the RTE withfixed scattering and absorption cross sections while the boundary conditionsvary; the other is when cross sections vary in a small domain and the boundaryconditions change many times. The solver can be decomposed into offline/onlinestages in these two cases. One only needs to calculate the offline stage onceand update the online stage when the parameters vary. Our proposed solver ismuch cheaper when one needs to solve RTE with multiple right-hand sides or whenthe cross sections vary in a small domain, thus can accelerate the speed ofsolving inverse RTE problems. We illustrate the online/offline decompositionbased on the Tailored Finite Point Method (TFPM), which is APAL on generalquadrilateral meshes.

A fast offline/online forward solver for stationary transport equation with multiple inflow boundary conditions and varying coefficients