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.
Many naturally-occuring models in the sciences are well-approximated by simplified models, using multiscale techniques. In such settings it is natural to ask about the relationship between inverse problems defined by the original problem and by the multiscale approximation. We develop an approach to this problem and exemplify it in the context of optical tomographic imaging.Optical tomographic imaging is a technique for infering the properties of biological tissue via measurements of the incoming and outgoing light intensity; it may be used as a medical imaging methodology. Mathematically, light propagation is modeled by the radiative transfer equation (RTE), and optical tomography amounts to reconstructing the scattering and the absorption coefficients in the RTE from boundary measurements. We study this problem in the Bayesian framework, focussing on the strong scattering regime. In this regime the forward RTE is close to the diffusion equation (DE). We study the RTE in the asymptotic regime where the forward problem approaches the DE, and prove convergence of the inverse RTE to the inverse DE in both nonlinear and linear settings. Convergence is proved by studying the distance between the two posterior distributions using the Hellinger metric, and using Kullback-Leibler divergence. * Submitted to the editors DATE.
For an overdetermined system Ax ≈ b with A and b given, the least-square (LS) formulation minx Ax − b 2 is often used to find an acceptable solution x. The cost of solving this problem depends on the dimensions of A, which are large in many practical instances. This cost can be reduced by the use of random sketching, in which we choose a matrix S with many fewer rows than A and b, and solve the sketched LS problem minx S(Ax − b) 2 to obtain an approximate solution to the original LS problem. Significant theoretical and practical progress has been made in the last decade in designing the appropriate structure and distribution for the sketching matrix S. When A and b arise from discretizations of a PDE-based inverse problem, tensor structure is often present in A and b. For reasons of practical efficiency, S should be designed to have a structure consistent with that of A. Can we claim similar approximation properties for the solution of the sketched LS problem with structured S as for fully-random S? We give estimates that relate the quality of the solution of the sketched LS problem to the size of the structured sketching matrices, for two different structures. Our results are among the first known for random sketching matrices whose structure is suitable for use in PDE inverse problems.
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