This thesis considers three inverse problems originating in mathematical imaging, remote sensing and simulation. First, it addresses a challenging image processing task: recovering images corrupted by multiplicative noise. Motivated by the success of multiscale hierarchical decomposition methods (MHDM) in image processing, a variety of both classical and new multiplicative noise removing models are adapted to the MHDM form. On the basis of previous work, tight and refined extensions of the multiplicative MHDM process are proposed. Existence and uniqueness of solutions for the proposed models is discussed, in addition to convergence properties. Moreover, the work introduces a discrepancy principle stopping criterion which prevents recovering excess noise in the multiscale reconstruction. The validity of all the proposed models is qualitatively and quantitatively evaluated through comprehensive numerical experiments and comparisons across several images degraded by multiplicative noise. By construction, these multiplicative multiscale hierarchical decomposition methods have the added benefit of recovering many scales of an image, which can provide features of interest beyond image denoising.
Second, this work considers an applied mathematical imaging problem in remote sensing. Accurate estimation of atmospheric wind velocity plays an important role in weather forecasting, flight safety assessment and cyclone tracking. Atmospheric data captured by infrared and microwave satellite instruments provide global coverage for weather analysis. Extracting wind velocity fields from such data has traditionally been done through feature tracking, correlation/matching or optical flow means from computer vision. However, these recover either sparse velocity estimates, oversmooth details or are designed for quasi-rigid body motions which over-penalize vorticity and divergence within the often turbulent weather systems. This thesis proposes a texture based optical flow procedure tailored for water vapor data. The method implements an $L1$ data term and total variation regularizer and employs a structure-texture image decomposition to identify key features which improve recoveries and help preserve the salient vorticity and divergence structures.
Based on the success of texture-features at improving the flow estimation, this procedure is extended to a multi-fidelity scheme which incorporates additional image features into the optical flow scheme. Both flow estimation methods are tested on simulated over-ocean mesoscale convective systems and convective and extratropical cyclone datasets, each of which have accompanying ground truth wind velocities allowing quantitative comparisons of each method's performance against existing optical flow methods.
Third, the thesis considers a data-driven reduced-order modeling problem. It is well known that full-resolution simulations of turbulence is computationally expensive (and often intractable), motivating the development of reduced-order models for practical engineering and science applications. However, developing accurate reduced models for turbulent flow is impeded by the loss of unresolved scales which critically influence the full state. The Mori-Zwanzig formalism is a mathematically exact strategy for performing reduced-order modeling in which reduced-order variables (observables) are evolved by memory kernels and an orthogonal dynamics term. Recently, frameworks for extracting Mori-Zwanzig kernels from simulated data have shown promise for reduced turbulence models. However, these same works highlight the importance of choosing an appropriate reduced-order observables set, and how this choice can substantially affect the model's overall success. This thesis proposes a joint-learning process to discover an improved set of observables in tandem with extracting the Mori-Zwanzig memory kernels. The process is tested on simulated two-dimensional turbulence data, and results show that the discovered set of observables enhances the Mori-Zwanzig based turbulence model's capabilities at predicting turbulent structures and statistics within the resolved space, both in short and long trajectories.
