Numerical algorithms for inverse problems in acoustics and optics

The objective of this dissertation is to develop computational algorithms for solving inverse coefficient problems for partial differential equations that appear in two medical imaging modalities. The aim of these inverse problems is to reconstruct optical properties of scattering media, such as biological tissues, from measured data collected on the surface of the media. In the first part of the dissertation, we study an inverse boundary value problems for the radiative transport equation. This inverse problem plays important roles in optics-based medical imaging techniques such as diffuse optical tomography and fluorescence optical tomography. We propose a robust reconstruction method that is based on subspace minimization techniques. The method splits the unknowns, both the unknown coefficient and the corresponding transport solutions (or a functional of it) into low-frequency and high-frequency components, and uses singular value decomposition to analytically recover part of low-frequency information. Minimization is then applied to recover part of the high-frequency components of the unknowns. We present some numerical simulations with synthetic data to demonstrate the performance of the proposed algorithm. In the second part of the dissertation, we develop a three-dimensional reconstruction algorithm for photoacoustic tomography in isotropic elastic media. There have been extensive study of photoacoustic tomography in recent years. However, all existing numerical reconstructions are developed for acoustic media in which case the model for wave propagation is the acoustic wave equation. We develop here a two-step reconstruction algorithm to reconstruct quantitatively optical properties, mainly the absorption coefficient and the Gr"uneisen coefficient using measured elastic wave data. The algorithm consists of an inverse source step where we reconstruct the source function in the elastic wave equation from boundary data and an inverse coefficient step where we reconstruct the coefficients of the diffusion equation using the result of the previous step as interior data. We present some numerical reconstruction results with synthetic data to demonstrate the performance of our algorithm. This is, to the best of our knowledge, the first reconstruction algorithm developed for quantitative photoacoustic imaging in elastic media. Despite the fact that we separate the dissertation into these two different parts to make each part more focused, the algorithms we developed in the two parts are closely related. In fact, if we replace the diffusion model for light propagation in photoacoustic imaging by the radiative transport model, which is often done in the literature, the algorithm we developed in the first part can be integrated into the algorithm in the second part after some minor modifications.