Inverse source problems for time-dependent radiative transport
Acosta Valenzuela, Sebastian
Borcea, Liliana; Riviere, Beatrice M.
Doctor of Philosophy
In the first part of this thesis, I develop a time reversal method for the radiative transport equation to solve two problems: an inverse problem for the recovery of an initial condition from boundary measurements, and the exact boundary controllability of the transport field with finite steering time. Absorbing and scattering effects, modeled by coefficients with low regularity, are incorporated in the formulation of these problems. This time reversal approach leads to a convergent iterative procedure to reconstruct the initial condition provided that the scattering coefficient is sufficiently small. Then, using duality arguments, I show that the solvability of the inverse problem leads to exact controllability of the transport field. The solution approach to both of these problems may have applications in areas such as optical imaging and optimization of radiation delivery. The second portion of the work is dedicated to the simultaneous recovery of a source of the form "s(t,x,d) f(x)" (with "s" known) and an isotropic initial condition "u0(x)", using the single measurement induced by these data. This result is part of an effort to reconstruct optical properties using unknown illumination embedded in the unknown medium. More precisely, based on exact boundary controllability, I derive a system of equations for the unknown terms "f" and "u0". The system is shown to be of Fredholm type if "s" satisfies a certain positivity condition. This condition requires that the radiation visits the region over which "f" is to be recovered. I show that for generic term "s" and weakly absorbing media, the inverse problem is well-posed.
Radiation transfer; Optical imaging; Tomography; Linear Boltzmann equation; Inverse problems