A Low-Rank Schwarz Method for Radiative Transfer Equation With Heterogeneous Scattering Coefficient

2021 ◽  
Vol 19 (2) ◽  
pp. 775-801
Author(s):  
Ke Chen ◽  
Qin Li ◽  
Jianfeng Lu ◽  
Stephen J. Wright
Keyword(s):  
Radiative Transfer ◽  
Transfer Equation ◽  
Radiative Transfer Equation ◽  
Scattering Coefficient ◽  
Low Rank ◽  
Schwarz Method

scholarly journals Diffusion Equation-Assisted Markov Chain Monte Carlo Methods for the Inverse Radiative Transfer Equation

Entropy ◽  
2019 ◽  
Vol 21 (3) ◽  
pp. 291 ◽  
Author(s):  
Qin Li ◽  
Kit Newton
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Radiative Transfer ◽  
Diffusion Equation ◽  
Posterior Distribution ◽  
Transfer Equation ◽  
Radiative Transfer Equation ◽  
Optical Tomography ◽  
Scattering Coefficient

Optical tomography is the process of reconstructing the optical properties of biological tissue using measurements of incoming and outgoing light intensity at the tissue boundary. Mathematically, light propagation is modeled by the radiative transfer equation (RTE), and optical tomography amounts to reconstructing the scattering coefficient in the RTE using the boundary measurements. In the strong scattering regime, the RTE is asymptotically equivalent to the diffusion equation (DE), and the inverse problem becomes reconstructing the diffusion coefficient using Dirichlet and Neumann data on the boundary. We study this problem in the Bayesian framework, meaning that we examine the posterior distribution of the scattering coefficient after the measurements have been taken. However, sampling from this distribution is computationally expensive, since to evaluate each Markov Chain Monte Carlo (MCMC) sample, one needs to run the RTE solvers multiple times. We therefore propose the DE-assisted two-level MCMC technique, in which bad samples are filtered out using DE solvers that are significantly cheaper than RTE solvers. This allows us to make sampling from the RTE posterior distribution computationally feasible.

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