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

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.

Dynamical Sampling with Langevin Normalization Flows

“…Since the transformation function is invertible, the integral of KL(q|p) can be calculated precisely. However, for NFL, the transformation functions demonstrated in Equations (12) and (13) are only approximately invertible because of the usage of the gradient information of the target distribution. Since the precise value of KL(q|p) cannot be obtained through integration, Monte Carlo estimation is used to calculate KL(q|p).…”

“…where f −1 θ is the inverse transformation function which can be calculated through Equation (12) and Equation (13). Since the update of x 1:D is divided into two parts, the calculation of ln det…”

“…where y 1:d , t 1 and t 2 can be calculated through Equations (12) and (13). However, in the progress of optimizing Equation (17), we find that the KL divergence may not be strictly non-negative because of the Monte Carlo integral, so we introduce a new objective function to overcome this problem.…”

“…Recent MCMC methods can be divided into two aspects. One class of the MCMC methods is slice sampling [11] and the other one is dynamical sampling [12,13]. The main problem of the slice sampler is that when sampling from the distributions with high dimensions, solving the slice interval can be very difficult.…”

MADFU: An Improved Malicious Application Detection Method Based on Features Uncertainty

“…The basic idea is to construct a Markov chain. Make the stable distribution of the Markov chain as the posterior distribution of the parameters to be estimated [ 27 ]. Monte Carlo integration is performed on samples with posterior distribution [ 28 ].…”