Inverse problems in transport and diffusion theory with applications in optical tomography

The work in this thesis mainly concerns inverse problems in transport and diffusion theory with an emphasis on applications in imaging techniques such as optical tomography and atmospheric remote sensing. Mathematically, inverse problems here involve the reconstruction of coefficients in partial differential (and integro-differential) equations from boundary measurements.; The first half of the thesis are devoted to the analysis and numerical solutions of inverse transport problems in optical tomography and atmospheric remote sensing. We developed two reconstruction algorithms for optical tomography in which we use the frequency domain transport equation as the forward model of light propagation in tissues. We show by numerical examples that the usage of the frequency domain information allows us to reduce the crosstalk between absorption and scattering coefficients in transport reconstructions from boundary current measurements. The crosstalk is much severe when steady-state data are used in the reconstruction. We have also analyzed an inverse problem related to the scattering-free atmospheric radiative transport equation. The inverse problem aims at reconstructing the concentration profiles of atmospheric gases (parameterized as functions of altitude in both the coefficient and the source term of the transport equation) from wavenumber-dependent boundary radiation measurement taken by space-borne infrared spectrometer. We showed in simplified situations that although the problem does admit a unique solution, it is severely ill-posed. We proposed an explicit procedure based on asymptotic analysis to reconstruct localized structures in the profile.; Modeling microscopic transport processes by macroscopic diffusion equations has its advantage many applications. Mathematically the modeling problem corresponds to the derivation of diffusion equations from transport equations. The second half of the thesis is devoted to such modeling problems and inverse problems related to them. We first compared in detail numerical reconstructions based the transport and diffusion equations in highly scattering and low absorbing media of small size. We characterized quantitatively the effect of inaccuracy in the diffusion approximation on the quality of the reconstructions. We then derived a generalized diffusion approximation for light propagation in highly diffusive media with extended thin non-scattering regions based on several previously reported results. We modeled those non-scattering extended regions by co-dimension one surfaces and used localized surface conditions to account for the effects of those non-scattering regions. Numerical simulations confirmed the accuracy of the new diffusion approximation. An inverse problem related to this generalized diffusion equation was then analyzed. The aim of this inverse problem is to reconstruct the locations of those extended non-scattering regions. We showed by numerical simulation that those regions be reconstructed from over-determined boundary measurements. The reconstruction method is based on shape sensitivity analysis and the level set method.