dissertationWe first study the inverse problem of recovering a complex Schro ?dinger potential from a discrete set of measurements of the solution to the Schro ?dinger equation using different source terms. We solve this problem by generalizing the inverse Born series method to nonlinear mappings between Banach spaces. In this general setting, we show convergence and stability of inverse Born series follow from a single problem- specific bound. We show this bound for the inverse Schro ?dinger problem, and study numerically an application of this inverse problem to transient hydraulic tomography. Additionally, we develop a family of iterative methods based on truncated inverse Born series that are akin to iterative methods based on truncated Taylor series. Next, we study the inverse problem of imaging scatterers in a homogeneous medium when only intensities of wavefields can be measured. Classic imaging meth- ods, such as Kirchhoff migration, rely on phase information contained in full waveform data and thus cannot be used directly with intensity-only data. In situations where scattered wavefields are small compared to the incident wavefields, we can form and solve a linear least squares problem to recover a projection (on a known subspace) of full waveform data from intensity data. We show that for sufficiently high frequencies, this projection gives a Kirchhoff image asymptotically equivalent to the Kirchhoff image obtained from full waveform data. We also generalize this imaging method to using stochastic incident fields with autocorrelation measurements. Finally, we study a mathematical model of grain growth in polycrystalline mate- rials. We review a simplified 1D grain growth model and an entropy-based theory for the evolution of an important statistic harvested from this model, the GBCD. The theory suggests the GBCD evolves according to a Fokker-Planck equation, which we validate numerically. We derive methods to estimate times from the GBCD, thus fitting it to Fokker-Planck time scales. This allows for direct comparisons of the GBCD with the Fokker-Planck solution, where we find qualitative agreement. We alsofind an energy dissipation identity which Fokker-Planck solutions must satisfy. We verify the GBCD satisfies this identity both qualitatively and quantitatively, further validating the Fokker-Planck model of GBCD evolution
