In the realm of discrete ill-posed problems, image deblurring is a challenging problem aimed at restoring clear and visually appealing images from their blurred counterparts. Over the years, various numerical techniques have been developed to solve this problem, each offering unique approaches to tackle blurring and noise.This thesis studies multilevel methods using Daubechies wavelets and Tikhonov regularization. The Daubechies wavelets are a family of orthogonal wavelets widely used in various fields because of their orthogonality and compact support. They have been widely applied in signal processing, image compression, and other applications. One key aspect of this investigation involves a comprehensive comparative analysis with Krylov methods, well-established iterative methods known for their efficiency and accuracy in solving large-scale inverse problems. The focus is on two well-known Krylov methods, namely hybrid LSQR and hybrid generalized minimal residual method \linebreak(GMRES). By contrasting the multilevel and Krylov methods, the aim is to discern their strengths and limitations, facilitating a deeper understanding of their applicability in diverse image-deblurring scenarios. Other critical comparison factors are the noise level adopted during the deblurring process and the amount of blur. To gauge their robustness and performance under different blurry and noisy conditions, this work explores how each method behaves with different noise levels from mild to severe and different amounts of blur from small to large. Moreover, this thesis combines multilevel and Krylov methods to test a new method for solving inverse problems. This work aims to provide valuable insights into the strengths and weaknesses of these multilevel Krylov methods by shedding light on their efficacy. Ultimately, the findings could have implications across diverse domains, including medical imaging, remote sensing, and multimedia applications, where high-quality and noise-free images are indispensable for accurate analysis and interpretation.

Deforestation in the Amazon rainforest has the potential to have devastating effects on ecosystems on both a local and global scale, making it one of the most environmentally threatening phenomena occurring today. In order to minimize deforestation in the Ama- zon and its consequences, it is helpful to analyze its occurrence using machine learning architectures such as the U-Net. The U-Net is a type of Fully Convolutional Network that has shown significant capability in performing semantic segmentation. It is built upon a symmetric series of downsampling and upsampling layers that propagate feature infor- mation into higher spatial resolutions, allowing for the precise identification of features on the pixel scale. Such an architecture is well-suited for identifying features in satellite imagery. In this thesis, we construct and train a U-Net to identify deforested areas in satellite imagery of the Amazon through semantic segmentation.

Deforestation in the Amazon rainforest has the potential to have devastating effects on ecosystems on both a local and global scale, making it one of the most environmentally threatening phenomena occurring today. In order to minimize deforestation in the Amazon and its consequences, it is helpful to analyze its occurrence using machine learning architectures such as the U-Net. The U-Net is a type of Fully Convolutional Network that has shown significant capability in performing semantic segmentation. It is built upon a symmetric series of downsampling and upsampling layers that propagate feature information into higher spatial resolutions, allowing for the precise identification of features on the pixel scale. Such an architecture is well-suited for identifying features in satellite imagery. In this thesis, we construct and train a U-Net to identify deforested areas in satellite imagery of the Amazon through semantic segmentation.

The variable projection method has been developed as a powerful tool for solvingseparable nonlinear least squares problems. It has proven effective in cases where the underlying model consists of a linear combination of nonlinear functions, such as exponential functions. In this thesis, a modified version of the variable projection method to address a challenging semi-blind deconvolution problem involving mixed Gaussian kernels is employed. The aim is to recover the original signal accurately while estimating the mixed Gaussian kernel utilized during the convolution process. The numerical results obtained through the implementation of the proposed algo- rithm are presented. These results highlight the method’s ability to approximate the true signal successfully. However, accurately estimating the mixed Gaussian kernel remains a challenging task. The implementation details, specifically focusing on con- structing a simplified Jacobian for the Gauss-Newton method, are explored. This contribution enhances the understanding and practicality of the approach.

This thesis addresses the problem of approximating analytic functions over general and compact multidimensional domains. Although the methods we explore can be used in complex domains, most of the tests are performed on the interval $[-1,1]$ and the square $[-1,1]\times[-1,1]$. Using Fourier and polynomial frame approximations on an extended domain, well-conditioned methods can be formulated. In particular, these methods provide exponential decay of the error down to a finite but user-controlled tolerance $\epsilon>0$. Additionally, this thesis explores two implementations of the frame approximation: a singular value decomposition (SVD)-regularized least-squares fit as described by Adcock and Shadrin in 2022, and a column and row selection method that leverages QR factorizations to reduce the data needed in the approximation. Moreover, strategies to reduce the complexity of the approximation problem by exploiting randomized linear algebra in low-rank algorithms are also explored, including the AZ algorithm described by Coppe and Huybrechs in 2020.

During the inversion of discrete linear systems, noise in data can be amplified and result in meaningless solutions. To combat this effect, characteristics of solutions that are considered desirable are mathematically implemented during inversion. This is a process called regularization. The influence of the provided prior information is controlled by the introduction of non-negative regularization parameter(s). Many methods are available for both the selection of appropriate regularization parame- ters and the inversion of the discrete linear system. Generally, for a single problem there is just one regularization parameter. Here, a learning approach is considered to identify a single regularization parameter based on the use of multiple data sets de- scribed by a linear system with a common model matrix. The situation with multiple regularization parameters that weight different spectral components of the solution is considered as well. To obtain these multiple parameters, standard methods are modified for identifying the optimal regularization parameters. Modifications of the unbiased predictive risk estimation, generalized cross validation, and the discrepancy principle are derived for finding spectral windowing regularization parameters. These estimators are extended for finding the regularization parameters when multiple data sets with common system matrices are available. Statistical analysis of these estima- tors is conducted for real and complex transformations of data. It is demonstrated that spectral windowing regularization parameters can be learned from these new esti- mators applied for multiple data and with multiple windows. Numerical experiments evaluating these new methods demonstrate that these modified methods, which do not require the use of true data for learning regularization parameters, are effective and efficient, and perform comparably to a supervised learning method based on es- timating the parameters using true data. The theoretical developments are validated for one and two dimensional image deblurring. It is verified that the obtained estimates of spectral windowing regularization parameters can be used effectively on validation data sets that are separate from the training data, and do not require known data.

A pneumonia-like illness emerged late in 2019 (coined COVID-19), caused by SARSCoV-2, causing a devastating global pandemic on a scale never before seen sincethe 1918/1919 influenza pandemic. This dissertation contributes in providing deeper qualitative insights into the transmission dynamics and control of the disease in the United States. A basic mathematical model, which incorporates the key pertinent epidemiological features of SARS-CoV-2 and fitted using observed COVID-19 data, was designed and used to assess the population-level impacts of vaccination and face mask usage in mitigating the burden of the pandemic in the United States. Conditions for the existence and asymptotic stability of the various equilibria of the model were derived. The model was shown to undergo a vaccine-induced backward bifurcation when the associated reproduction number is less than one. Conditions for achieving vaccine-derived herd immunity were derived for three of the four FDA-approved vaccines (namely Pfizer, Moderna and Johnson & Johnson vaccine), and the vaccination coverage level needed to achieve it decreases with increasing coverage of moderately and highly-effective face masks. It was also shown that using face masks as a singular intervention strategy could lead to the elimination of the pandemic if moderate or highly-effective masks are prioritized and pandemic elimination prospects are greatly enhanced if the vaccination program is combined with a face mask use strategy that emphasizes the use of moderate to highly-effective masks with at least moderate coverage. The model was extended in Chapter 3 to allow for the assessment of the impacts of waning and boosting of vaccine-derived and natural immunity against the BA.1 Omicron variant of SARS-CoV-2. It was shown that vaccine-derived herd immunity can be achieved in the United States via a vaccination-boosting strategy which entails fully vaccinating at least 72% of the susceptible populace. Boosting of vaccine-derived immunity was shown to be more beneficial than boosting of natural immunity. Overall, this study showed that the prospects of the elimination of the pandemic in the United States were highly promising using the two intervention measures.

Magnetic liquids called ferrofluids have been used in applications ranging from audio speaker cooling and rotary pressure seals to retinal detachment surgery and implantable artificial glaucoma valves. Recently, ferrofluids have been investigated as a material for use in magnetically controllable liquid droplet robotics. Liquid droplet robotics is an emerging technology that aims to apply control theory to manipulate fluid droplets as robotic agents to perform a wide range of tasks. Furthermore, magnetically controlled micro-robotics is another popular area of study where manipulating a magnetic field allows for the control of magnetized micro-robots. Both of these emerging fields have potential for impact toward medical applications: liquid characteristics such as being able to dissolve various compounds, be injected via a needle, and the potential for the human body to automatically filter and remove a liquid droplet robot, make liquid droplet robots advantageous for medical applications; while the ability to remotely control the torques and forces on an untethered microrobot via modulating the magnetic field and gradient is also highly advantageous. The research described in this dissertation explores applications and methods for the electromagnetic control of ferrofluid droplet robots. First, basic electrical components built from fluidic channels containing ferrofluid are made remotely tunable via the placement of ferrofluid within the channel. Second, a ferrofluid droplet is shown to be fully controllable in position, stretch direction, and stretch length in two dimensions using proportional-integral-derivative (PID) controllers. Third, control of a ferrofluid’s position, stretch direction, and stretch length is extended to three dimensions, and control gains are optimized via a Bayesian optimization process to achieve higher accuracy. Finally, magnetic control of both single and multiple ferrofluid droplets in two dimensions is investigated via a visual model predictive control approach based on machine learning. These achievements take both liquid droplet robotics and magnetic micro-robotics fields several steps closer toward real-world medical applications such as embedded soft electronic health monitors, liquid-droplet-robot-based drug delivery, and automated magnetically actuated surgeries.

Solving partial differential equations on surfaces has many applications including modeling chemical diffusion, pattern formation, geophysics and texture mapping. This dissertation presents two techniques for solving time dependent partial differential equations on various surfaces using the partition of unity method. A novel spectral cubed sphere method that utilizes the windowed Fourier technique is presented and used for both approximating functions on spherical domains and solving partial differential equations. The spectral cubed sphere method is applied to solve the transport equation as well as the diffusion equation on the unit sphere. The second approach is a partition of unity method with local radial basis function approximations. This technique is also used to explore the effect of the node distribution as it is well known that node choice plays an important role in the accuracy and stability of an approximation. A greedy algorithm is implemented to generate good interpolation nodes using the column pivoting QR factorization. The partition of unity radial basis function method is applied to solve the diffusion equation on the sphere as well as a system of reaction-diffusion equations on multiple surfaces including the surface of a red blood cell, a torus, and the Stanford bunny. Accuracy and stability of both methods are investigated.