A Fast Offline/Online Forward Solver for Stationary Transport Equation with Multiple Inflow Boundary Conditions and Varying Coefficients (Communications in Computational Physics, 2025)
It is of great interest to solve the inverse problem of stationary radiativetransport equation (RTE) in optical tomography. The standard way is to formulate theinverse problem into an optimization problem, but the bottleneck is that one has tosolve the forward problem repeatedly, which is time-consuming. Due to the opticalproperty of biological tissue, in real applications, optical thin and thick regions coexistand are adjacent to each other, and the geometry can be complex. To use coarse meshesand save the computational cost, the forward solver has to be asymptotic preservingacross 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 fixedscattering and absorption cross sections while the boundary conditions vary; the otheris when cross sections vary in a small domain and the boundary conditions changemany times. The solver can be decomposed into offline/online stages in these twocases. One only needs to calculate the offline stage once and update the online stagewhen the parameters vary. Our proposed solver is much cheaper when one needs tosolve RTE with multiple right-hand sides or when the cross sections vary in a smalldomain, thus can accelerate the speed of solving inverse RTE problems. We illustratethe online/offline decomposition based on the Tailored Finite Point Method (TFPM),which is APAL on general quadrilateral meshes.