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<div>This thesis compiles my work on three projects.</div><div>The first project studies the cancellation of singularities in the inversion of two X-ray type transforms in the presence of conjugate points. The second project studies the recovery of singularities for the weighted cone transform. The third project studies the phenomenon of Rayleigh waves and Stoneley waves in the isotropic elastic wave equation of variable coefficients with a curved boundary.</div>

In the need of computational tools for epsilon-multiplicity, we provide a criterion for a module with a rank E inside a free module F to have rational epsilon-multiplicity in terms of the finite generation of the saturation Rees algebra of E. In this case, the multiplicity can be related to a Hilbert multiplicity of certain graded algebra. A particular example of this situation is provided: it is shown that the epsilon-multiplicity of monomial modules is Noetherian. Numerical evidence is provided that leads to a conjecture formula for the epsilon-multiplicity of certain monomial curves in the 3-affine space.

In this thesis, I study two problems. First, I generalize a result by H-Y Chen to show that if $X$ is a smooth variety of general type and irregularity $q\geq 1$ that embeds into its Albanese variety as a smooth variety $Y$ of general type with codimension one or two, then $|Aut(X)|\leq |Aut(F_{min})||Aut(Y)|$ where $F_{min}$ is the minimal model of a general fiber. Then I describe a special type of fibration called a K-Fibration as a generalization to Kodaira Fibrations where we can compute its Chern numbers in dimensions 2 and 3. K-Fibrations act as an initial step in constructing examples of varieties that satisfy the generalization with the goal of computing their automorphism group explicitly.

<p>High-dimensional regression problems are ubiquitous in science and engineering. Deep learning has been a critical tool for solving a wide range of high-dimensional problems with surprising performance. Even though in theory neural networks have good properties in terms of approximation and optimization, numerically obtaining an accurate neural network solution is a challenging problem due to the highly non-convex objective function and implicit bias of least square optimization. In this dissertation, we mainly discuss two topics involving the high dimensional regression using efficient deep learning algorithms. These two topics include solving PDEs with high dimensional domains and data-driven dynamical modeling. </p> <p><br></p> <p>In the first topic, we aim to develop an efficient solver for PDE problems. Firstly, we focus on neural network structures to increase efficiency. We propose a data-driven activation function called reproducing activation function which can reproduce traditional approximation tools and enable faster convergence of deep neural network training with smaller parameter cost. Secondly, we target the application of neural networks to mitigate the numerical issues that hamper the traditional approach. As an example, we develop a neural network solver for elliptic PDEs on unknown manifolds and verify its effectiveness for the large-scale problem. </p> <p><br></p> <p>In the second topic, we aim to enhance the accuracy of learning the dynamical system from data by incorporating the prior. In the missing dynamics problem, taking advantage of known partial dynamics, we propose a framework that approximates a map that takes the memories of the resolved and identifiable unresolved variables to the missing components in the resolved dynamics. With this framework, we achieve a low error to predict the missing component, enabling the accurate prediction of the resolved variables. In the recovering Hamiltonian dynamics, by the energy conservation property, we learn the conserved Hamiltonian function instead of its associated vector field. To better learn the Hamiltonian from the stiff dynamics, we identify and splits the</p> <p>training data into stiff and nonstiff portions, and adopt different learning strategies based on the classification to reduce the training error. </p>

<div>Deep learning has become an important toolkit for data science and artificial intelligence. In contrast to its practical success across a wide range of fields, theoretical understanding of the principles behind the success of deep learning has been an issue of controversy. Optimization, as an important component of theoretical machine learning, has attracted much attention. The optimization problems induced from deep learning is often non-convex and</div><div>non-smooth, which is challenging to locate the global optima. However, in practice, global convergence of first-order methods like gradient descent can be guaranteed for deep neural networks. In particular, gradient descent yields zero training loss in polynomial time for deep neural networks despite its non-convex nature. Besides that, another mysterious phenomenon is the compelling performance of Deep Residual Network (ResNet). Not only</div><div>does training ResNet require weaker conditions, the employment of residual connections by ResNet even enables first-order methods to train the neural networks with an order of magnitude more layers. Advantages arising from the usage of residual connections remain to be discovered.</div><div><br></div><div>In this thesis, we demystify these two phenomena accordingly. Firstly, we contribute to further understanding of gradient descent. The core of our analysis is the neural tangent hierarchy (NTH) that captures the gradient descent dynamics of deep neural networks. A recent work introduced the Neural Tangent Kernel (NTK) and proved that the limiting</div><div>NTK describes the asymptotic behavior of neural networks trained by gradient descent in the infinite width limit. The NTH outperforms the NTK in two ways: (i) It can directly study the time variation of NTK for neural networks. (ii) It improves the result to non-asymptotic settings. Moreover, by applying NTH to ResNet with smooth and Lipschitz activation function, we reduce the requirement on the layer width m with respect to the number of training samples n from quartic to cubic, obtaining a state-of-the-art result. Secondly, we extend our scope of analysis to structural properties of deep neural networks. By making fair and consistent comparisons between fully-connected network and ResNet, we suggest strongly that the particular skip-connection architecture possessed by ResNet is the main</div><div>reason for its triumph over fully-connected network.</div>

<p>Applications in materials and biological imaging are currently limited by the ability to collect high-resolution data over large areas in practical amounts of time. One possible solution to this problem is to collect low-resolution data and apply a super-resolution interpolation algorithm to produce a high-resolution image. However, state-of-the-art super-resolution algorithms are typically designed for natural images, require aligned pairing of high and low-resolution training data for optimal performance, and do not directly incorporate a data-fidelity mechanism.</p><p><br></p><p>We present a Multi-Resolution Data Fusion (MDF) algorithm for accurate interpolation of low-resolution SEM and TEM data by factors of 4x and 8x. This MDF interpolation algorithm achieves these high rates of interpolation by first learning an accurate prior model denoiser for the TEM sample from small quantities of unpaired high-resolution data and then balancing this learned denoiser with a novel mismatched proximal map that maintains fidelity to measured data. The method is based on Multi-Agent Consensus Equilibrium (MACE), a generalization of the Plug-and-Play method, and allows for interpolation at arbitrary resolutions without retraining. We present electron microscopy results at 4x and 8x super resolution that exhibit reduced artifacts relative to existing methods while maintaining fidelity to acquired data and accurately resolving sub-pixel-scale features.</p>

<p>Many kernel matrices from differential equations or data science applications possess low or approximately low off-diagonal rank for certain key matrix subblocks; such matrices are referred to as rank-structured. Operations on rank-structured matrices like factorization and linear system solution can be greatly accelerated by converting them into hierarchical matrix forms, such as the hiearchically semiseparable (HSS) matrix form. The dominant cost of this conversion process, called HSS construction, is the low-rank approximation of certain matrix blocks. Low-rank approximation is also a required step in many other contexts throughout numerical linear algebra. In this work, a proxy point low-rank approximation method is detailed for general analytic kernel matrices, in both one and several dimensions. A new accuracy analysis for this approximation is also provided, as well as numerical evidence of its accuracy. The extension of this method to kernels in several dimensions is novel, and its new accuracy analysis makes it a convenient choice to use over existing proxy point methods. Finally, a new HSS construction algorithm using this method for certain Cauchy and Toeplitz matrices is given, which is asymptotically faster than existing methods. Numerical evidence for the accuracy and efficacy of the new construction algorithm is also provided.</p>

<div>Red blood cells (RBCs) make up 40-45% of blood and play an important role in oxygen transport. That transport depends on the RBC distribution throughout the body, which is highly heterogeneous. That distribution, in turn, depends on how RBCs are distributed or partitioned at diverging vessel bifurcations where one vessel flows into two. Several studies have used mathematical modeling to consider RBC partitioning at such bifurcations in order to produce useful insights. However, these studies assume that the vessel wall is a flat impenetrable homogeneous surface. While this is a good first approximation, especially for larger vessels, the vessel wall is typically coated by a flexible, porous endothelial surface layer (ESL) that is 0.5-1 microns thick. To better understand the possible effects of this layer on RBC partitioning, a diverging capillary bifurcation is analyzed using a flexible, two-dimensional RBC model. The model is also used to investigate RBC deformation and penetration of the ESL region when ESL properties are varied. The RBC is represented using interconnected viscoelastic elements. Stokes flow equations (viscous flow) model the surrounding fluid. The flow in the ESL is modeled using the Brinkman approximation for porous media with a corresponding hydraulic resistivity. The resistance of the ESL to compression is modeled using an osmotic pressure difference. The study includes isolated cells that pass through the bifurcation one at a time with no cell-cell interactions and two cells that pass through the bifurcation at the same time and interact with each other. A range of physiologically relevant hydraulic resistivities and osmotic pressure differences are explored.</div><div><br></div><div>For isolated cell simulations, decreasing hydraulic resistivity and/or decreasing osmotic pressure difference produced four behaviors: 1) RBC distribution nonuniformity increased; 2) RBC deformation decreased; 3) RBCs slowed down slightly; and 4) RBCs penetrated more deeply into the ESL. The presence of an altered flow profile and the ESL's resistance to penetration were primary factors responsible for these behaviors. In certain scenarios, ESL penetration was deep enough to present a possibility of cell adhesion, as can occur in pathological situations.</div><div><br></div><div>For paired cell simulations, more significant and complex changes were observed. Three types of effects that alter partitioning as hydraulic resistivity is changed are identified. Decreasing hydraulic resistivity in the ESL produced lower RBC deformation. Including cell-cell interactions tended to increase deformation sharply compared to isolated cell scenarios. ESL penetration generally decreased for lower hydraulic resistivities except in scenarios with significant cell-cell interactions. This was primarily due to changes in flow profiles induced by the altered hydraulic resistivity levels.</div>

This thesis studies problems in computational topology through the lens of semi-algebraic geometry. We first give an algorithm for computing a semi-algebraic basis for the first homology group, H1(S,F), with coefficients in a field F, of any given semi-algebraic set S⊂Rk defined by a closed formula. The complexity of the algorithm is bounded singly exponentially. More precisely, if the given quantifier-free formula involves s polynomials whose degrees are bounded by d, the complexity of the algorithm is bounded by (sd)^kO⁽¹⁾.This algorithm generalizes well known algorithms having singly exponential complexity for computing a semi-algebraic basis of the zero-th homology group of semi-algebraic sets, which is equivalent to the problem of computing a set of points meeting every semi-algebraically connected component of the given semi-algebraic set at a unique point. We then turn our attention to the Reeb graph, a tool from Morse theory which has recently found use in applied topology due to its ability to track the changes in connectivity of level sets of a function. The roadmap of a set, a construction that arises in semi-algebraic geometry, is a one-dimensional set that encodes information about the connected components of a set. In this thesis, we show that the Reeb graph and, more generally, the Reeb space, of a semi-algebraic set is homeomorphic to a semi-algebraic set, which opens up the algorithmic problem of computing a semi-algebraic description of the Reeb graph. We present an algorithm with singly-exponential complexity that realizes the Reeb graph of a function f:X→Y as a semi-algebraic quotient using the roadmap of X with respect to f.