We present several graph-based algorithms for image processing and classification of high- dimensional data. The first (semi-supervised) method uses a graph adaptation of the classical numerical Merriman-Bence-Osher (MBO) scheme, and can be extended to the multiclass case via the Gibbs simplex. We show examples of the application of the algorithm in the areas of image inpainting (both binary and grayscale), image segmentation and classification on benchmark data sets. We have also applied this algorithm to the problem of object detection using hyperspectral video sequences as a data set. In addition, a second related model is introduced. It uses a diffuse interface model based on the Ginzburg-Landau functional, related to total variation compressed sensing and image processing. A multiclass extension is introduced using the Gibbs simplex, with the functional’s double-well potential modified to handle the multiclass case. The version minimizes the functional using a convex splitting numerical scheme. In our computations, we make use of fast numerical solvers for finding the eigenvectors and eigenvalues of the graph Laplacian, and take advantage of the sparsity of the matrix. Experiments on benchmark data sets show that our models produce results that are comparable with or outperform the state-of-the-art algorithms. The second (semi-supervised) method develops a global minimization framework for binary classification of high-dimensional data. It combines recent convex optimization methods for image processing with recent graph based variational models for data segmentation. Two convex splitting algorithms are proposed, where graph-based PDE techniques are used to solve some of the subproblems. It is shown that global minimizers can be guaranteed for semi-supervised segmentation with two regions. If constraints on the volume of the regions are incorporated, global minimizers cannot be guaranteed, but can often be obtained in practice and otherwise be closely approximated. We perform a thorough comparison to recent MBO (Merriman-Bence-Osher) and phase field methods, and show the advantage of the proposed algorithms. Lastly, we present the current work (unsupervised method) related to normalized cuts. The method uses a clever alternative to the normalized cut to solve the binary classification problem. In particular, we work with the Ginzburg-Landau functional. In addition, we use a generalized graphical framework, so several different graph Laplacians are tested and their results are compared.

In recent years, a massive expansion in the amount of available network data in fields such as social networks, food networks in ecology, similarity networks in machine learning, transportation networks, brain networks, and many others has motivated the development of "network science'' to describe all this data. One of the fundamental branches in network science is "community detection,'' or the decomposition of large networks into coherent subnetworks, which is useful for visualization, data exploration, hypothesis formation, approximation of network dynamics, link prediction, and a host of other tasks.

Two of the most well-known frameworks for community detection are modularity optimization and stochastic block modeling. They can often uncover meaningful community structure in networks from diverse applications. However, both of these approaches are computationally demanding. In this dissertation, I will show how these two statistically-motivated frameworks for community detection can be reinterpreted more geometrically using the language of graph total variation (TV) and (discretized) surface tension, respectively. This change in perspective allows one to leverage algorithms and analytical tools developed for other problems in the fields of compressed sensing, materials science, and nonlinear partial differential equations. One also can adapt arguments from other domains to obtain theoretical guarantees on the performance of these algorithms. I illustrate these approaches on a number of synthetic and real-world datasets, yielding results competitive with other state-of-the-art techniques on problems from machine learning, image processing, social networks, and biological networks.

A thin liquid film flowing down a vertical fiber exhibits complex and captivating interfacial dynamics of an unsteady flow, including the formation of droplets, irregular traveling waves, and a string of beads. The dynamics of traveling beads are particularly applicable in various fluid experiments due to their high surface-area-to-volume ratio, which enhances gas diffusion towards the liquid surface. Recent studies verified that when the flow undergoes regime transitions, the motion of the film thickness changes dramatically. The dynamics of the liquid film are influenced by the cylindrical geometry of the fiber and the presence of stabilizing and destabilizing forces, which pose additional challenges for numerical simulations. Many current numerical simulations of fiber coating dynamics, such as direct numerical simulation (DNS), rely on the full Navier-Stokes equations. Such numerical simulations incur significant computational costs, often requiring several days on a desktop computer. As a result of the high computational cost, it is not feasible to simulate regime transitions or extend the computational domain further downstream of the fluid.

In order to overcome these challenges, several reduced-order models using lubrication approximations have been developed. These models are much simpler than the full Navier-Stokes equations, yet they are capable of capturing droplet dynamics and transient patterns of the flow. However, these models have drawbacks in terms of their versatility, as one needs to modify the form of the equations depending on the assumptions made regarding boundary conditions and the scales of the problem. Depending on the boundary conditions at the solid-liquid interface, one may observe singularities, cusps, and non-classical shocks in finite time. If one were to consider a liquid film with a moderate thickness or a moderate Reynolds number, the resulting equation would exhibit significant differences.

In this dissertation, we consider a model for fiber coating at low Reynolds numbers with geometry such that the fluid thickness is larger than the fiber radius. We present a computationally efficient numerical method that can maintain the positivity of the film thickness as well as conserve the volume of the fluid in a coarse mesh setting. Our method allows simulations of regimes with isolated droplets and the Rayleigh-Plateau instability, commonly observed in laboratory experiments but particularly difficult to simulate. We create our positivity-preserving numerical method in the following way. First, we present a conserved and dissipative quantity for our continuous model. Next, we construct a continuous in time and discrete-in-space numerical method that satisfies the discrete equivalent of conservation of mass and an entropy estimate. We provide a proof of the positivity of our numerical method using a priori and a posteriori bounds and a proof of second-order consistency. Finally, we show that our method can be implemented efficiently using an adaptive time-stepping method to describe solutions that correspond quantitatively to laboratory experiments.

Many accurate spatiotemporal data sets have recently become available for research. Real-world applications create strong demands for a better multivariate point-process modeling. In this thesis, we develop new multivariate models with generalization ability and scalability.

The first two chapters provide a research background, real-world problems and a mathematical introduction to point-process models.

In chapter 3, we develop a nonparametric method for multivariate spatiotemporal Hawkes processes with applications on network reconstruction. In contrast to prior work, which has often focused on exclusively temporal information, our approach uses spatiotemporal information and does not assume a specific parametric form. Our results demonstrate that, in comparison to using only temporal data, our approach yields improved network reconstruction, providing a basis for meaningful subsequent analysis---such as examinations of community structure and motifs---of the reconstructed networks.

In chapter 4, we present a fast and accurate estimation method for multivariate Hawkes processes. Our method, with guaranteed consistency, combines two estimation approaches. Extensive numerical experiments, with synthetic data and real-world social network data, show that our method improves the accuracy, scalability and computational efficiency of prevailing estimation approaches. Moreover, it greatly boosts the performance of Hawkes process-based models on social network reconstruction and helps to understand the spatiotemporal triggering dynamics over social media.

In chapter 5, we focus on multivariate spatial point processes, which can describe heterotopic data over space. However, highly multivariate intensities are computationally challenging due to the curse of dimensionality. To bridge this gap, we introduce a declustering-based hidden-variable model that leads to an efficient inference via a variational autoencoder (VAE). We also prove that this model is a generalization of the VAE-based model for collaborative filtering. This leads to an interesting application of spatial point-process models to recommender systems. Experimental results show the method's utility on both synthetic data and real-world data.

Finally, in chapter 6, we show how multivariate point processes can be applied to opioid overdose events and real-time prediction of the hourly crime rate. In chapter 7, we discuss future directions and conclude the thesis.

Microfluidics, fluid dynamics on the micro-scale, has been an active area of research due to its applications - separating particles, mixing fluids, and generating microdroplets. This thesis presents a mathematical model and analysis of two microfluidic technologies. The first is the use of micro-particles to form uniform microdroplets. We build a model using surface tension energy minimization to prove that the uniform distribution of droplets is an energy minimum distribution. Additionally, we use a random pairwise interaction model to understand the amount of mixing needed to achieve the uniform distribution. Based on this mathematical model we show that adding larger particles reduces the amount of mixing required and suggests an improved particle design.

The second is the focusing of particles flowing through a duct. Depending on both the size and shape of the particle and duct, the focusing can happen on different positions in the duct cross-section or not happen at all. Understanding the particle dynamics of the cross-section is important for the development of related technologies. We develop a simplified approximate model that combines the drag force due to Dean flow and the lift force with a single parameter that characterizes the relative strength between the two. The cross-sectional dynamics of this model preserve that of the full model. The simplified model exhibits three distinct dynamics as the single parameter changes. We analyze the three cross-sectional dynamics and the bifurcations between them.

We present several results on the subject of graph-based semi-supervised learning and a novel application of network analysis to analyzing complex spatiotemporal data. The first piece of work showcases a specific graph-based semi-supervised learning algorithm in the application to ego-activity classification in body-worn video. The classification method is inspired by three interrelated processes: the Allen-Cahn equation, the Merriman-Bence-Osher scheme, and mean curvature flow. We present results on real-world body-worn videos and demonstrate our method's comparable performance to supervised methods. The second piece of work presents semi-supervised learning problem in the framework of Bayesian inverse problems; we prove posterior consistency and elucidate how hyperparameter choices in the Bayesian model combine to affect the contraction rates of the posterior. The third piece of work presents a method of uncertainty quantification in the aforementioned framework; we also develop the foundations for a system with a human in the loop who serves to provide additional class labels based on the uncertainty quantification. The fourth piece of work further extends the Bayesian inverse problem framework to the active learning problem. We introduce an adaptation of non-Gaussian Bayesian models to allow efficient calculations previously done only on Gaussian models and a novel way of choosing new training data. The last piece of work presents a multivariate point-process model that infers latent relationships from complex spatiotemporal data.

In this dissertation, a blob method of measuring spatial coverage by a swarm of agents is presented, and two models are introduced: a macroscopic model, consisting of a system of Advection-Diffusion-Reaction equations that govern the spatial distribution of the swarm at the population level; and a microscopic model, represented by a stochastic differential equation, describing the individual behavior of each agent in the swarm. Depending on different tasks, multiple control frameworks are proposed to drive the swarm to a target distribution, and are proved to be both valid in theory and robust in real world case.

First, we briefly review the history of Advection-Diffusion-Reaction (ADR) system in the literature. The ADR system models the agents' motion as drifted brownian motion, and the agents' reaction as probabilistic transition between different states. The key idea of the ADR model lies in the fact that the green's function of Advection-Diffusion Equation is the conditional probability density function of the corresponding drifted brownian motion. The proposed point mass approach verifies that the error between the macroscopic and the microscopic model converges with respect to the inverse of the square root of the swarm size.

Next, we explores a stochastic approach for controlling swarms of independent robots toward a target distribution in a bounded domain. The robots are resource-constraint: they lack both

communication and localization capabilities, and can only gather local information by measuring a scalar field (e.g. concentration of a chemical) from the environment. A simple control law is introduced to govern the diffusion of each robot, so that the distribution of the swarm converges to a pre-defined target distribution over time. The key point behind the control framework is that the solution of heat equation converges to the first-order eigenvector of the Laplacian operator. We further confirm the robustness of the control law in real world case by conducting simulations in computer and test bed experiments in multiple cases of target distributions, where the swarm achieves the theorized convergence to the target distribution despite deviations from assumptions underpinning the theory.

Finally, we presents a novel procedure for computing parameters of a robotic swarm that guarantee coverage performance by the swarm within a specified error from a target spatial distribution. The main contribution is the analysis of the dependence of this error on two key parameters: the number of robots in the swarm and the robot sensing radius, in which we view each robot as a blob instead of a point mass. We model the population dynamics of the swarm as an advection-diffusion-reaction partial differential equation (PDE) with time-dependent advection and reaction terms. We derive rigorous bounds on the discrepancies between the target distribution and the coverage achieved by individual-based and PDE models of the swarm. We use these bounds to select the swarm size that will achieve coverage performance within a given error and the corresponding robot sensing radius that will minimize this error. We also apply the optimal control approach to compute the robots' velocity field and task-switching rates. We validate our procedure through simulations of a scenario in which a robotic swarm must achieve a specified density of pollination activity over a crop field.

Graph-based machine learning is a powerful framework for analyzing and understanding complex data structures in various domains. This thesis introduces novel graph-based methods in multiple image analysis tasks, including classification, segmentation, and unmixing, as well as their application in enhancing large language models. The key contributions include: (1) the development of new core-set selection and batch active learning methods that significantly improve the efficiency of graph-based active learning while maintaining its effectiveness; (2) the integration of graph learning, active learning, and advanced feature embedding methods to construct pipelines for SAR image classification and multi- or hyperspectral image segmentation, outperforming neural network-based classifiers or segmenters in semi-supervised learning tasks with limited training data; (3) the incorporation of graph-based regularization into the optimization problem of hyperspectral unmixing, enabling the utilization of a small amount of labeled pixels to greatly improve the performance compared to blind unmixing; and (4) the extension of graph Laplacian-based methods to automatically construct knowledge graphs in combination with large language models, enhancing their information retrieval and response generation capabilities.

The proposed methods showcase the effectiveness and versatility of graph-based approaches in addressing challenges such as limited labeled data, computational efficiency, and knowledge representation. The thesis demonstrates the potential of graph-based methods in pushing the boundaries of image and document analysis and their applicability in a wide range of machine learning problems.

The demand for analyzing patterns and structures of data is growing dramatically in recent years. The study of network structure is pervasive in sociology, biology, computer science, and many other disciplines. My research focuses on network and high-dimensional data analysis, using graph based models and tools from sparse optimization. The specific question about networks we are studying is "clustering": partitioning a network into cohesive groups. Depending on the contexts, these tightly connected groups can be social units, functional modules, or components of an image.

My work consists of both theoretical analysis and numerical simulation. We first analyze some social network and image datasets using a quality function called "modularity", which is a popular model for clustering in network science. Then we further study the modularity function from a novel perspective: with my collaborators we reformulate modularity optimization as a minimization problem of an energy functional that consists of a total variation term and an L2 balance term. By employing numerical techniques from image processing and L1 compressive sensing, such as the Merriman-Bence-Osher (MBO) scheme, we develop a variational algorithm for the minimization problem.

Along a similar line of research, we work on a multi-class segmentation problem using the piecewise constant Mumford-Shah model in a graph setting. We propose an efficient algorithm for the graph version of Mumford-Shah model using the MBO scheme. Theoretical analysis is developed and a Lyapunov functional is proven to decrease as the algorithm proceeds. Furthermore, to reduce the computational cost for large datasets, we incorporate the Nystrom extension method to efficiently approximates eigenvectors of the graph Laplacian based on a small portion of the weight matrix. Finally, we implement the proposed method on the problem of chemical plume detection in hyper-spectral video data. These graph based clustering algorithms we proposed improve the time efficiency significantly for large scale datasets. In the last chapter, we also propose an incremental reseeding strategy for clustering, which is an easy-to-implement and highly parallelizable algorithm for multiway graph partitioning. We demonstrate experimentally that this algorithm achieves state-of-the-art performance in terms of cluster purity on standard benchmark datasets. Moreover, the algorithm runs an order of magnitude faster than the other algorithms.