Title:
Mathematical Approaches to Identification problems -- Counting, RNA folding, and PDE identification
Mathematical Approaches to Identification problems -- Counting, RNA folding, and PDE identification
Author(s)
Tang Rajchel, Mengyi
Advisor(s)
Kang, Sung Ha
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Abstract
Mathematical 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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Date Issued
2023-11-28
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Resource Type
Text
Resource Subtype
Dissertation