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.

When solving analysis, estimation, and control problems for Partial Differential Equations (PDEs) via computational methods, one must resolve three main challenges: (a) the lack of a universal parametric representation of PDEs; (b) handling unbounded differential operators that appear as parameters; and (c), enforcing auxiliary constraints such as Boundary conditions and continuity conditions. To address these challenges, an alternative representation of PDEs called the `Partial Integral Equation' (PIE) representation is proposed in this work. Primarily, the PIE representation alleviates the problem of the lack of a universal parametrization of PDEs since PIEs have, at most, $12$ Partial Integral (PI) operators as parameters. Naturally, this also resolves the challenges in handling unbounded operators because PI operators are bounded linear operators. Furthermore, for admissible PDEs, the PIE representation is unique and has no auxiliary constraints --- resolving the last of the $3$ main challenges. The PIE representation for a PDE is obtained by finding a unique unitary map from the states of the PIE to the states of the PDE. This map shows a PDE and its associated PIE have equivalent system properties, including well-posedness, internal stability, and I/O behavior. Furthermore, this unique map also allows us to construct a well-defined dual representation that can be used to solve optimal control problems for a PDE. Using the equivalent PIE representation of a PDE, mathematical and computational tools are developed to solve standard problems in Control theory for PDEs. In particular, problems such as a test for internal stability, Input-to-Output (I/O) $L_2$-gain, $\hinf$-optimal state observer design, and $\hinf$-optimal full state-feedback controller design are solved using convex-optimization and Lyapunov methods for linear PDEs in one spatial dimension. Once the PIE associated with a PDE is obtained, Lyapunov functions (or storage functions) are parametrized by positive PI operators to obtain a solvable convex formulation of the above-stated control problems. Lastly, the methods proposed here are applied to various PDE systems to demonstrate the application.

The main objective of this work is to study novel stochastic modeling applications to cybersecurity aspects across three dimensions: Loss, attack, and detection. First, motivated by recent spatial stochastic models with cyber insurance applications, the first and second moments of the size of a typical cluster of bond percolation on finite graphs are studied. More precisely, having a finite graph where edges are independently open with the same probability $p$ and a vertex $x$ chosen uniformly at random, the goal is to find the first and second moments of the number of vertices in the cluster of open edges containing $x$. Exact expressions for the first and second moments of the size distribution of a bond percolation cluster on essential building blocks of hybrid graphs: the ring, the path, the random star, and regular graphs are derived. Upper bounds for the moments are obtained by using a coupling argument to compare the percolation model with branching processes when the graph is the random rooted tree with a given offspring distribution and a given finite radius. Second, the Petri Net modeling framework for performance analysis is well established; extensions provide enough flexibility to examine the behavior of a permissioned blockchain platform in the context of an ongoing cyberattack via simulation. The relationship between system performance and cyberattack configuration is analyzed. The simulations vary the blockchain's parameters and network structure, revealing the factors that contribute positively or negatively to a Sybil attack through the performance impact of the system. Lastly, the denoising diffusion probabilistic models (DDPM) ability for synthetic tabular data augmentation is studied. DDPMs surpass generative adversarial networks in improving computer vision classification tasks and image generation, for example, stable diffusion. Recent research and open-source implementations point to a strong quality of synthetic tabular data generation for classification and regression tasks. Unfortunately, the present state of literature concerning tabular data augmentation with DDPM for classification is lacking. Further, cyber datasets commonly have highly unbalanced distributions complicating training. Synthetic tabular data augmentation is investigated with cyber datasets and performance of well-known metrics in machine learning classification tasks improve with augmentation and balancing.

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 work presents a thorough analysis of reconstruction of global wave fields (governed by the inhomogeneous wave equation and the Maxwell vector wave equation) from sensor time series data of the wave field. Three major problems are considered. First, an analysis of circumstances under which wave fields can be fully reconstructed from a network of fixed-location sensors is presented. It is proven that, in many cases, wave fields can be fully reconstructed from a single sensor, but that such reconstructions can be sensitive to small perturbations in sensor placement. Generally, multiple sensors are necessary. The next problem considered is how to obtain a global approximation of an electromagnetic wave field in the presence of an amplifying noisy current density from sensor time series data. This type of noise, described in terms of a cylindrical Wiener process, creates a nonequilibrium system, derived from Maxwell’s equations, where variance increases with time. In this noisy system, longer observation times do not generally provide more accurate estimates of the field coefficients. The mean squared error of the estimates can be decomposed into a sum of the squared bias and the variance. As the observation time $\tau$ increases, the bias decreases as $\mathcal{O}(1/\tau)$ but the variance increases as $\mathcal{O}(\tau)$. The contrasting time scales imply the existence of an ``optimal'' observing time (the bias-variance tradeoff). An iterative algorithm is developed to construct global approximations of the electric field using the optimal observing times. Lastly, the effect of sensor acceleration is considered. When the sensor location is fixed, measurements of wave fields composed of plane waves are almost periodic and so can be written in terms of a standard Fourier basis. When the sensor is accelerating, the resulting time series is no longer almost periodic. This phenomenon is related to the Doppler effect, where a time transformation must be performed to obtain the frequency and amplitude information from the time series data. To obtain frequency and amplitude information from accelerating sensor time series data in a general inhomogeneous medium, a randomized algorithm is presented. The algorithm is analyzed and example wave fields are reconstructed.

Balancing temporal shortages of renewable energy with natural gas for the generation of electricity is a challenge for dispatchers. This is compounded by the recent proposal of blending cleanly-produced hydrogen into natural gas networks. To introduce the concepts of gas flow, this thesis begins by linearizing the partial differential equations (PDEs) that govern the flow of natural gas in a single pipe. The solution of the linearized PDEs is used to investigate wave attenuation and characterize critical operating regions where linearization is applicable. The nonlinear PDEs for a single gas are extended to mixtures of gases with the addition of a PDE that governs the conservation of composition. The gas mixture formulation is developed for general gas networks that can inject or withdraw arbitrary time-varying mixtures of gases into or from the network at arbitrarily specified nodes, while being influenced by time-varying control actions of compressor units. The PDE formulation is discretized in space to form a nonlinear control system of ordinary differential equations (ODEs), which is used to prove that homogeneous mixtures are well-behaved and heterogeneous mixtures may be ill-behaved in the sense of monotone-ordering of solutions. Numerical simulations are performed to compute interfaces that delimit monotone and periodic system responses. The ODE system is used as the constraints of an optimal control problem (OCP) to minimize the expended energy of compressors. Moreover, the ODE system for the natural gas network is linearized and used as the constraints of a linear OCP. The OCPs are digitally implemented as optimization problems following the discretization of the time domain. The optimization problems are applied to pipelines and small test networks. Some qualitative and computational applications, including linearization error analysis and transient responses, are also investigated.

Ferrofluidic microrobots have emerged as promising tools for minimally invasive medical procedures, leveraging their unique properties to navigate through complex fluids and reach otherwise inaccessible regions of the human body, thereby enabling new applications in areas such as targeted drug delivery, tissue engineering, and diagnostics. This dissertation develops a model-predictive controller for the external magnetic manipulation of ferrofluid microrobots. Several experiments are performed to illustrate the adaptability and generalizability of the control algorithm to changes in system parameters, including the three-dimensional reference trajectory, the velocity of the workspace fluid, and the size, orientation, deformation, and velocity of the microrobotic droplet. A linear time-invariant control system governing the dynamics of locomotion is derived and used as the constraints of a least squares optimal control algorithm to minimize the projected error between the actual trajectory and the desired trajectory of the microrobot. The optimal control problem is implemented after time discretization using quadratic programming. In addition to demonstrating generalizability and adaptability, the accuracy of the control algorithm is analyzed for several different types of experiments. The experiments are performed in a workspace with a static surrounding fluid and extended to a workspace with fluid flowing through it. The results suggest that the proposed control algorithm could enable new capabilities for ferrofluidic microrobots, opening up new opportunities for applications in minimally invasive medical procedures, lab-on-a-chip, and microfluidics.

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.

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.