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Applied Math Seminar
A fast solver for radiative transport equation and applications in
optical imaging
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I will present an efficient forward solver for steady-state or frequency-domain radiative transfer equation (RTE) on 2D and 3D structured and unstructured meshes with vacuum boundary condition or reflection boundary condition. Due to high dimensionality the key issue is to construct an efficient iterative scheme that can deal with various behaviors of the solution in different regimes, such as transport, diffusion, optical thick, and forward peaking . In our algorithm we use a direct angular discretization and upwind type of spatial discretization that preserves properties of both scattering and differential operators of RTE. To solve the large linear system after discretization, we construct an efficient iterative scheme based on Gauss-Seidel and proper angular dependent ordering. With this iterative scheme as the relaxation we implement various multigrid methods in both angular and physical space. Our algorithm can deal with different scattering regimes efficiently. Efficiency and accuracy of our algorithm is demonstrated by comparison with both analytical solutions and Monte Carlo solutions, and various numerical tests in optical imaging. Based on this algorithm, a multi-level imaging algorithm is developed for optical tomorgraphy. If time permits, I will also talk about recent work on compressive sensing (CS) in bioluminescence tomography (BLT) based on RTE. We show that direct use of Green's function of RTE as the basis and standard measurement data for reconstruction will not satisfy the restricted isometry property (RIP). We propose an algorithm to transform the standard setup into a new system that satisfies the RIP. Hence, with compressive sensing method, we produce much improved results over standard reconstruction method. |