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

Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53705, USA
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Entropy 2019, 21(3), 291; https://doi.org/10.3390/e21030291
Received: 18 December 2018 / Revised: 28 February 2019 / Accepted: 8 March 2019 / Published: 18 March 2019
(This article belongs to the Special Issue Information Theory and Stochastics for Multiscale Nonlinear Systems)
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Abstract

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. View Full-Text
Keywords: multi-level MCMC; radiative transfer equation; inverse problems; diffusion limit multi-level MCMC; radiative transfer equation; inverse problems; diffusion limit
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Li, Q.; Newton, K. Diffusion Equation-Assisted Markov Chain Monte Carlo Methods for the Inverse Radiative Transfer Equation. Entropy 2019, 21, 291.

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