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SUMMARYWe present stochastic projection schemes for approximating the solution of a class of deterministic linear elliptic partial differential equations defined on random domains. The key idea is to carry out spatial discretization using a combination of finite element methods and stochastic mesh representations. We prove a result to establish the conditions that the input uncertainty model must satisfy to ensure the validity of the stochastic mesh representation and hence the well posedness of the problem. Finite element spatial discretization of the governing equations using a stochastic mesh representation results in a linear random algebraic system of equations in a polynomial chaos basis whose coefficients of expansion can be non-intrusively computed either at the element or the global level. The resulting randomly parametrized algebraic equations are solved using stochastic projection schemes to approximate the response statistics. The proposed approach is demonstrated for modeling diffusion in a square domain with a rough wall and heat transfer analysis of a three-dimensional gas turbine blade model with uncertainty in the cooling core geometry. The numerical results are compared against Monte-Carlo simulations, and it is shown that the proposed approach provides high-quality approximations for the first two statistical moments at modest computational effort.

We develop discontinuous Galerkin framework for solving direct and inverse problems in fluorescence diffusion optical tomography in turbid media. We show the advantages and the disadvantages of this method by comparing it with previously developed framework based on the finite volume discretization. The reconstruction algorithm was used with time-gated experimental dataset acquired by imaging a highly scattering cylindrical phantom concealing small fluorescent tubes. Optical parameters, quantum yield and lifetime were simultaneously reconstructed. Reconstruction results are presented and discussed.

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SUMMARYWe propose a discontinuous Galerkin discretization scheme for the forward modelling in optical diffusion tomography in highly scattering media to facilitate dynamic mesh adaptation for complex domains. In addition, the numerical method is also shown to effectively deal with inhomogeneities in optical properties and refractive index mismatch at material interfaces. The accuracy of the method is demonstrated over a model concentric spherical layers problem where the discontinuous Galerkin method is compared against an analytical solution.