Fast Algorithm For Solving Time-dependent Multiscale radiative transport Equation
This work addresses the need for faster solvers in time-dependent radiative transport, which is important for applications like medical imaging and atmospheric science.
The paper tackles the high computational cost of solving time-dependent radiative transport equations by combining an adaptive tailored finite point scheme for spatial discretization with the Recursive Skeleton Method for explicit multilevel decomposition of the inverse operator, achieving high accuracy and efficiency across diverse scenarios.
When solving the time-dependent radiative transport equation (RTE), implicit time discretization is often employed for its robustness and stability. This results in a sequence of steady-state RTEs with identical cross-sections but varying source terms, whose repeated solution is computationally costly. To address this, we first apply the adaptive tailored finite point scheme (TFPS) for spatial discretization. This scheme exploits prior knowledge of the background media's optical properties to adaptively compress the angular domain, constructing a compressed linear system. A key feature is its ability to reconstruct the layer structure after compression, faithfully capturing the variance at the layer. We then use the Recursive Skeleton Method (RSM) to obtain an explicit multilevel decomposition of the inverse discrete operator, which is reused for all steady-state solutions. Numerical experiments show that our framework achieves high accuracy and significant efficiency across diverse scenarios.