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.

Amyotrophic Lateral Sclerosis (ALS) is a fatal neurodegenerative disease characterized by the progressive loss of motor function. Pathological mechanisms and clinical measures vary extensively from patient to patient, creating a spectrum of disease phenotypes with a poorly understood influence on individual outcomes like disease duration. The inability to ascertain patient phenotype has hindered clinical trial design and the development of more personalized and effective therapeutics. Wholistic analytical methods (‘-omics’) have provided unprecedented molecular resolution into cellular and system level disease processes and offer a foundation to better understand ALS disease variability. Building off initiatives by the New York Genome Center ALS Consortium and Target ALS groups, the goal of this work was to stratify a large patient cohort utilizing a range of bioinformatic strategies and bulk tissue gene expression (transcriptomes) from the brain and spinal cord. Central Hypothesis: Variability in the onset and progression of ALS is partially captured by molecular subgroups (subtypes) with distinct gene expression profiles and implicated pathologies. Work presented in this dissertation addresses the following: (Chapter 2): The use of unsupervised clustering and gene enrichment methods for the identification and characterization of patient subtypes in the postmortem cortex and spinal cord. Results obtained from this Chapter establish three ALS subtypes, identify uniquely dysregulated pathways, and examine intra-patient concordance between regions of the central nervous system. (Chapter 3): Patient subtypes from Chapter 2 are considered in the context of clinical outcomes, leveraging multiple survival models and gene co-expression analyses. Results from this Chapter establish a weak association between ALS subtype and clinical outcomes including disease duration and age at symptom onset. (Chapter 4): Utilizing differential expression analysis, ‘marker’ genes are defined and leveraged with supervised classification (“machine learning”) methods to develop a suite of classifiers design to stratify patients by subtype. Results from this Chapter provide postmortem marker genes for two of the three ALS subtypes and offer a foundation for clinical stratification. Significance: Knowledge gained from this research provides a foundation to stratify patients in the clinic and prior to enrollment in clinical trials, offering a path toward improved therapies.

Gene expression models are key to understanding and predicting transcriptional dynamics. This thesis devises a computational method which can efficiently explore a large, highly correlated parameter space, ultimately allowing the author to accurately deduce the underlying gene network model using discrete, stochastic mRNA counts derived through the non-invasive imaging method of single molecule fluorescence in situ hybridization (smFISH). An underlying gene network model consists of the number of gene states (distinguished by distinct production rates) and all associated kinetic rate parameters. In this thesis, the author constructs an algorithm based on Bayesian parametric and nonparametric theory, expanding the traditional single gene network inference tools. This expansion starts by increasing the efficiency of classic Markov-Chain Monte Carlo (MCMC) sampling by combining three schemes known in the Bayesian statistical computing community: 1) Adaptive Metropolis-Hastings (AMH), 2) Hamiltonian Monte Carlo (HMC), and 3) Parallel Tempering (PT). The aggregation of these three methods decreases the autocorrelation between sequential MCMC samples, reducing the number of samples required to gain an accurate representation of the posterior probability distribution. Second, by employing Bayesian nonparametric methods, the author is able to simultaneously evaluate discrete and continuous parameters, enabling the method to devise the structure of the gene network and all kinetic parameters, respectively. Due to the nature of Bayesian theory, uncertainty is evaluated for the gene network model in combination with the kinetic parameters. Tools brought from Bayesian nonparametric theory equip the method with an ability to sample from the posterior distribution of all possible gene network models without pre-defining the gene network structure, i.e. the number of gene states. The author verifies the method’s robustness through the use of synthetic snapshot data, designed to closely represent experimental smFISH data sets, across a range of gene network model structures, parameters and experimental settings (number of probed cells and timepoints).

As the impacts of climate change worsen in the coming decades, natural hazards are expected to increase in frequency and intensity, leading to increased loss and risk to human livelihood. The spatio-temporal statistical approaches developed and applied in this dissertation highlight the ways in which hazard data can be leveraged to understand loss trends, build forecasts, and study societal impacts of losses. Specifically, this work makes use of the Spatial Hazard Events and Losses Database which is an unparalleled source of loss data for the United States. The first portion of this dissertation develops accurate loss baselines that are crucial for mitigation planning, infrastructure investment, and risk communication. This is accomplished thorough a stationarity analysis of county level losses following a normalization procedure. A wide variety of studies employ loss data without addressing stationarity assumptions or the possibility for spurious regression. This work enables the statistically rigorous application of such loss time series to modeling applications. The second portion of this work develops a novel matrix variate dynamic factor model for spatio-temporal loss data stratified across multiple correlated hazards or perils. The developed model is employed to analyze and forecast losses from convective storms, which constitute some of the highest losses covered by insurers. Adopting factor-based approach, forecasts are achieved despite the complex and often unobserved underlying drivers of these losses. The developed methodology extends the literature on dynamic factor models to matrix variate time series. Specifically, a covariance structure is imposed that is well suited to spatio-temporal problems while significantly reducing model complexity. The model is fit via the EM algorithm and Kalman filter. The third and final part of this dissertation investigates the impact of compounding hazard events on state and regional migration in the United States. Any attempt to capture trends in climate related migration must account for the inherent uncertainties surrounding climate change, natural hazard occurrences, and socioeconomic factors. For this reason, I adopt a Bayesian modeling approach that enables the explicit estimation of the inherent uncertainty. This work can provide decision-makers with greater clarity regarding the extent of knowledge on climate trends.

This dissertation comprises two projects: (i) Multiple testing of local maxima for detection of peaks and change points with non-stationary noise, and (ii) Height distributions of critical points of smooth isotropic Gaussian fields: computations, simulations and asymptotics. The first project introduces a topological multiple testing method for one-dimensional domains to detect signals in the presence of non-stationary Gaussian noise. The approach involves conducting tests at local maxima based on two observation conditions: (i) the noise is smooth with unit variance and (ii) the noise is not smooth where kernel smoothing is applied to increase the signal-to-noise ratio (SNR). The smoothed signals are then standardized, which ensures that the variance of the new sequence's noise becomes one, making it possible to calculate $p$-values for all local maxima using random field theory. Assuming unimodal true signals with finite support and non-stationary Gaussian noise that can be repeatedly observed. The algorithm introduced in this work, demonstrates asymptotic strong control of the False Discovery Rate (FDR) and power consistency as the number of sequence repetitions and signal strength increase. Simulations indicate that FDR levels can also be controlled under non-asymptotic conditions with finite repetitions. The application of this algorithm to change point detection also guarantees FDR control and power consistency. The second project focuses on investigating the explicit and asymptotic height densities of critical points of smooth isotropic Gaussian random fields on both Euclidean space and spheres.The formulae are based on characterizing the distribution of the Hessian of the Gaussian field using the Gaussian orthogonally invariant (GOI) matrices and the Gaussian orthogonal ensemble (GOE) matrices, which are special cases of GOI matrices. However, as the dimension increases, calculating explicit formulae becomes computationally challenging. The project includes two simulation methods for these distributions. Additionally, asymptotic distributions are obtained by utilizing the asymptotic distribution of the eigenvalues (excluding the maximum eigenvalues) of the GOE matrix for large dimensions. However, when it comes to the maximum eigenvalue, the Tracy-Widom distribution is utilized. Simulation results demonstrate the close approximation between the asymptotic distribution and the real distribution when $N$ is sufficiently large.

\begin{abstract}The human immunodeficiency virus (HIV) pandemic, which causes the syndrome of opportunistic infections that characterize the late stage HIV disease, known as the acquired immunodeficiency syndrome (AIDS), remains a major public health challenge to many parts of the world. This dissertation contributes in providing deeper qualitative insights into the transmission dynamics and control of the HIV/AIDS disease in Men who have Sex with Men (MSM) community. A new mathematical model (which is relatively basic), which incorporates some of the pertinent aspects of HIV epidemiology and immunology and fitted using the yearly new case data of the MSM population from the State of Arizona, was designed and used to assess the population-level impact of awareness of HIV infection status and condom-based intervention, on the transmission dynamics and control of HIV/AIDS in an MSM community. Conditions for the existence and asymptotic stability of the various equilibria ofthe model were derived. The numerical simulations showed that the prospects for the effective control and/or elimination of HIV/AIDS in the MSM community in the United States are very promising using a condom-based intervention, provided the condom efficacy is high and the compliance is moderate enough. The model was extended in Chapter 3 to account for the effect of risk-structure, staged-progression property of HIV disease, and the use of pre-exposure prophylaxis (PrEP) on the spread and control of the disease. The model was shown to undergo a PrEP-induced \textit{backward bifurcation} when the associated control reproduction number is less than one. It was shown that when the compliance in PrEP usage is $50%(80%)$ then about $19.1%(34.2%)$ of the yearly new HIV/AIDS cases recorded at the peak will have been prevented, in comparison to the worst-case scenario where PrEP-based intervention is not implemented in the MSM community. It was also shown that the HIV pandemic elimination is possible from the MSM community even for the scenario when the effective contact rate is increased by 5-fold from its baseline value, if low-risk individuals take at least 15 years before they change their risky behavior and transition to the high-risk group (regardless of the value of the transition rate from high-risk to low-risk susceptible population).

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.

Tracking disease cases is an essential task in public health; however, tracking the number of cases of a disease may be difficult not every infection can be recorded by public health authorities. Notably, this may happen with whole country measles case reports, even such countries with robust registration systems. Eilertson et al. (2019) propose using a state-space model combined with maximum likelihood methods for estimating measles transmission. A Bayesian approach that uses particle Markov Chain Monte Carlo (pMCMC) is proposed to estimate the parameters of the non-linear state-space model developed in Eilertson et al. (2019) and similar previous studies. This dissertation illustrates the performance of this approach by calculating posterior estimates of the model parameters and predictions of the unobserved states in simulations and case studies. Also, Iteration Filtering (IF2) is used as a support method to verify the Bayesian estimation and to inform the selection of prior distributions. In the second half of the thesis, a birth-death process is proposed to model the unobserved population size of a disease vector. This model studies the effect of a disease vector population size on a second affected population. The second population follows a non-homogenous Poisson process when conditioned on the vector process with a transition rate given by a scaled version of the vector population. The observation model also measures a potential threshold event when the host species population size surpasses a certain level yielding a higher transmission rate. A maximum likelihood procedure is developed for this model, which combines particle filtering with the Minorize-Maximization (MM) algorithm and extends the work of Crawford et al. (2014).

A leading crisis in the United States is the opioid use disorder (OUD) epidemic. Opioid overdose deaths have been increasing, with over 100,000 deaths due to overdose from April 2020 to April 2021. This dissertation presents two mathematical models to address illicit OUD (IOUD), treatment, and recovery within an epidemiological framework. In the first model, individuals remain in the recovery class unless they relapse. Due to the limited availability of specialty treatment facilities for individuals with OUD, a saturation treat- ment function was incorporated. The second model is an extension of the first, where a casual user class and its corresponding specialty treatment class were added. Using U.S. population data, the data was scaled to a population of 200,000 to find parameter estimates. While the first model used the heroin-only dataset, the second model used both the heroin and all-illicit opioids datasets. Backward bifurcation was found in the first IOUD model for realistic parameter values. Additionally, bistability was observed in the second IOUD model with the heroin-only dataset. This result implies that it would be beneficial to increase the availability of treatment. An alarming effect was discovered about the high overdose death rate: by 2038, the disease-free equilibrium would be the only stable equilibrium. This consequence is concerning because although the goal is for the epidemic to end, it would be preferable to end it through treatment rather than overdose. The IOUD model with a casual user class, its sensitivity results, and the comparison of parameters for both datasets, showed the importance of not overlooking the influence that casual users have in driving the all-illicit opioid epidemic. Casual users stay in the casual user class longer and are not going to treatment as quickly as the users of the heroin epidemic. Another result was that the users of the all-illicit opioids were going to the recovered class by means other than specialty treatment. However, the relapse rates for those individuals were much more significant than in the heroin-only epidemic. The results above from analyzing these models may inform health and policy officials, leading to more effective treatment options and prevention efforts.

The high uncertainty of renewables introduces more dynamics to power systems. The conventional way of monitoring and controlling power systems is no longer reliable. New strategies are needed to ensure the stability and reliability of power systems. This work aims to assess the use of machine learning methods in analyzing data from renewable integrated power systems to aid the decisionmaking of electricity market participants. Specifically, the work studies the cases of electricity price forecast, solar panel detection, and how to constrain the machine learning methods to obey domain knowledge.Chapter 2 proposes to diversify the data source to ensure a more accurate electricity price forecast. Specifically, the proposed two-stage method, namely the rerouted method, learns two types of mapping rules: the mapping between the historical wind power and the historical price and the forecasting rule for wind generation. Based on the two rules, we forecast the price via the forecasted generation and the learned mapping between power and price. The massive numerical comparison gives guidance for choosing proper machine learning methods and proves the effectiveness of the proposed method. Chapter 3 proposes to integrate advanced data compression techniques into machine learning algorithms to either improve the predicting accuracy or accelerate the computation speed. New semi-supervised learning and one-class classification methods are proposed based on autoencoders to compress the data while refining the nonlinear data representation of human behavior and solar behavior. The numerical results show robust detection accuracy, laying down the foundation for managing distributed energy resources in distribution grids. Guidance is also provided to determine the proper machine learning methods for the solar detection problem. Chapter 4 proposes to integrate different types of domain knowledge-based constraints into basic neural networks to guide the model selection and enhance interpretability. A hybrid model is proposed to penalize derivatives and alter the structure to improve the performance of a neural network. We verify the performance improvement of introducing prior knowledge-based constraints on both synthetic and real data sets.