A fast offline/online forward solver for stationary transport equation with multiple inflow boundary conditions and varying coefficients
Jingyi Fu, Min Tang
TL;DR
The paper addresses accelerating the inverse stationary radiative transport equation in optical tomography by delivering a fast offline/online forward solver that is asymptotic-preserving across interfaces. It builds on the Tailored Finite Point Method (TFPM), leverages offline computation of small local systems, and enables rapid online updates when boundary data or cross sections vary. Two practical scenarios are analyzed: Case I with fixed cross sections and varying boundary conditions and Case II with cross sections varying in a small subdomain. Numerical results in 1D/2D demonstrate substantial online speedups versus standard solvers, and the approach is compatible with general quadrilateral meshes and preserves uniform accuracy near boundary/interface layers.
Abstract
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.