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ItemMathematical Approaches to Identification problems -- Counting, RNA folding, and PDE identification(Georgia Institute of Technology, 2023-11-28) Tang Rajchel, MengyiMathematical algorithms have become an essential tool in uncovering hidden patterns and unraveling dynamic behaviors within complex datasets, aiding in gaining deeper insights and making informed choices in an era driven by data-driven decision-making. This thesis introduces several numerical algorithms addressing identification problems derived from mathematical models. These works place a specific emphasis on identifying and predicting structures and patterns within various types of datasets while also offering the capacity to forecast the behavior of future data. Our contributions include StemP, a novel algorithm using graph notations predicting RNA sequence secondary structures with simplicity and being deterministic, without a training process. Additionally, our work Counting Objects by Diffused Index(CODI) efficiently counts objects in digital images using a diffusion algorithm with an operator-splitting approach and the alternating direction minimization method inspired by color inpainting, delivering results within seconds.Furthermore, our works WeakIdent and FourierIdent focus on identifying differential equations in the physical and frequency domains, respectively. WeakIdent provides a general and robust framework for identifying differential equations, enhancing accuracy with proposed innovative mechanisms of narrow-fit and trimming. FourierIdent explores the benefits and challenges of frequency domain utilization in differential equation identification, presenting comprehensive experiments to demonstrate their benefits in robustness over state-of-the-art methods.
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ItemA Novel Delay Differential Equation Model of the Germinal Center Reaction and an Algorithm for Minimum Length Surveillance Paths(Georgia Institute of Technology, 2022-05-19) Ide, BenjaminThe humoral adaptive immune system in vertebrates includes a process called the germinal center reaction in which B-cells rapidly increase their binding affinity to an antigen that is part of a pathogen. A fraction of germinal center B cells differentiates into plasma cells that secrete antibodies. Antibodies bind to the pathogen and neutralize it. In a secondary immune response to the same pathogen, memory B-cells and long-lived plasma cells generated during the primary immune response encode higher affinity antibodies and are able to fight the pathogen more efficiently. We develop a delay differential equation model of the germinal center reaction incorporating known physical mechanisms. We find that secondary immune responses including low affinity seeder B-cells outperform those seeded only with higher affinity. This helps to explain a recent laboratory observation that a high fraction of seeder B-cells in a secondary immune response are naive. Further, two mechanisms of antibody feedback are explored, where antibodies produced in the reaction interact with the reaction itself. Negative feedback occurs via epitope masking, which is consistent with experimental data. Positive feedback occurs via improved antigen presentation on follicular dendritic cells, which is a mechanism we propose. Additionally, we propose a novel path optimization algorithm. Given a path connected environment, our proposed algorithm finds the shortest paths from which surfaces in the environment are surveyed under a limited visibility constraint. We further explore how this is related to the inradius problem in classical geometry: What is the shortest curve whose convex hull includes the unit sphere? The solution is known for closed curves, but not for open curves. Our algorithm seems to converge numerically to the true solution for closed curves and to the best-known conjecture for open curves. This offers validation of our method and evidence for the open path conjecture.
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ItemOptimal Motion Planning and Computational Optimal Transport(Georgia Institute of Technology, 2022-04-28) Sun, HaodongIn many real life problems, the decision making process is guided by the principle of cost minimization. In this thesis, we focus on analyzing the theoretical properties and designing computational methods under the framework of several classical cost minimization problems, including optimal motion planning and optimal transport (OT). Over the past decades, motion planning has attracted large amount of attention in robotics. Given certain configurations in the environment, we are looking for trajectories which move the robot from one to the other. To produce high-quality trajectories, we propose a new computational method to design smooth and collision-free trajectories for motion planning with one or more robots. The functional cost in model leads to short and smooth trajectories. The designed method can also be generalized to problems with multiple robots. The idea of optimal transport naturally arises from many application scenarios including economy, computer science, etc. Optimal transport provides powerful tools for comparing probability measures in various types. However, obtaining the optimal transport plan is generally a computationally-expensive task. We start with an entropy transport problem as a relaxed version of original optimal transport problem with soft marginals, and propose an efficient algorithm to produce sample approximation for the optimal transport plan. This method can directly output samples from optimal plan between two continuous marginals without any discretization and network training. An inverse problem of OT is also of our interest. We present a computational framework for learning the cost function from the given optimal transport plan. The cost learning problem is reformulated as an unconstrained convex optimization problem and two efficient algorithms are proposed for discrete and continuous cost learning.
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ItemMathematical and Data-driven Pattern Representation with Applications in Image Processing, Computer Graphics, and Infinite Dimensional Dynamical Data Mining(Georgia Institute of Technology, 2021-04-30) He, YuchenPatterns represent the spatial or temporal regularities intrinsic to various phenomena in nature, society, art, and science. From rigid ones with well-defined generative rules to flexible ones implied by unstructured data, patterns can be assigned to a spectrum. On one extreme, patterns are completely described by algebraic systems where each individual pattern is obtained by repeatedly applying simple operations on primitive elements. On the other extreme, patterns are perceived as visual or frequency regularities without any prior knowledge of the underlying mechanisms. In this thesis, we aim at demonstrating some mathematical techniques for representing patterns traversing the aforementioned spectrum, which leads to qualitative analysis of the patterns' properties and quantitative prediction of the modeled behaviors from various perspectives. We investigate lattice patterns from material science, shape patterns from computer graphics, submanifold patterns encountered in point cloud processing, color perception patterns applied in underwater image processing, dynamic patterns from spatial-temporal data, and low-rank patterns exploited in medical image reconstruction. For different patterns and based on their dependence on structured or unstructured data, we present suitable mathematical representations using techniques ranging from group theory to deep neural networks.