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Topological data analysis emerged as a field in the 2000s and has proven very useful for examining the shape of data sets. Using persistent homology as their main methodology researchers has succesfully applied the theory presented in this paper to study as varied subjects as robot motion, brain connectivity, network theory, finger print analysis and computer vision.  The mathematical theory behind persistent homology has traditionally required training far beyond what an average engineer would have. Therefore much theory is usually left out of presentations meant for an audience outside of a mathematics department. This paper contains a novel approach to the presentation of this theory, maintaining mathematical rigour while only using linear algebra as its building blocks.

<p> Systems of differential-algebraic equations (DAEs) arise in numerious applications, and there has been considerable research on solving DAE initial value problems (IVPs). Existing methods and software for solving DAEs usually handle at most index-three problems. However, DAE problems of index three and higher do arise, for example, in actuator dynamics, multi-stage processes, and optimization.</p> <p> We present the method of J. Pryce and N. Nedialkov for solving DAEs, which can be of high index, fully implicit, and contain derivatives of order higher than one. We solve such DAEs by expanding their solution in Taylor series (TS). To compute Taylor coefficients, we employ J. Pryce's structural analysis and automatic differentiation. Then we compute an approximate TS solution with appropriate stepsize and project this solution to satisfy the constraints (explicit and hidden) of the problem.</p> <p> This thesis discusses the algorithms involved in this method, including the algorithms for Taylor coefficients computation, consistent point projection, error estimation, stepsize control, and the overall integration process. The author has implemented a software package named HIDAETS (High-Index DAE by Taylor Series). In this thesis, we present the specification, design, implementation, and usage of HIDAETS. Numerical results on several high-index DAEs are reported. These results demonstrate that HIDAETS is efficient and accurate for solving IVP in DAEs.</p> / Thesis / Master of Science (MSc)

<p> Systems of differential-algebraic equations (DAEs) arise in applications such as circuit simulation, models of chemical processes, optimal control, and multi-body dynamics. Informally, the index of a DAE is the number of differentiations needed to convert it to an ordinary differential equation. The index generally indicates the difficulty of solving a DAE problem. The higher the index of a DAE, the more difficult it is to solve it numerically.</p> <p> Structural index analysis plays a crucial role in solving DAE problems. In this thesis, we present two methods for index analysis, namely, Pryce's structural analysis (SA) and Linninger's symbolic-numeric (SN) analysis. We provide a Matlab tool implementing these two approaches: an Automatic Structural Index Analyzer (ASIA). We discuss the underlying algorithms, which include generating a signature matrix and computing SA index, computing a system Jacobian, and generating a symbolic-numeric matrix and computing SN index. We also present implementation issues and illustrate how ASIA is used.</p> <p> Numerical experiments show that ASIA can produce reliable structural information. We also show examples on which structural analysis fails, and how ASIA detects such situations.<p> / Thesis / Master of Science (MSc)

We propose two efficient automatic differentiation (AD) schemes to compute incrementally Taylor series and System Jacobian for solving differential-algebraic equations (DAEs) by Taylor series. Our schemes are based on topological ordering of a DAE's computational graph and then partitioning the topologically sorted nodes using structural information obtained from the DAE. Solving a DAE by Taylor series is carried out in stages. From one stage to another, partitions of the computational graph are incrementally activated so that we can reuse Taylor coefficients and gradients computed in previous stages. As a result, the computational complexity of evaluating a System Jacobian is independent of the number of stages. We also develop a common subexpression elimination (CSE) method to build a compact computational graph through operator overloading. The CSE method is of linear time complexity, which makes it suitable as a preprocessing step for general operator overloaded computing. By applying CSE, all successive overloaded computation can save time and memory. Furthermore, the computational graph of a DAE reveals its internal sparsity structure. Based on it, we devise an algorithm to propagate gradients in the forward mode of AD using compressed vectors. This algorithm can save both time and memory when computing the System Jacobian for sparse DAEs. We have integrated our approaches into the \daets solver. Computational results show multiple-fold speedups against two popular AD tools, \FAD~and ADOL-C, when solving various sparse and dense DAEs. / Thesis / Master of Science (MSc)

Persistent homology is a technique of topological data analysis that seeks to understand the shape of data. We study the effectiveness of a single-layer perceptron and gradient boosted classification trees in classifying perhaps the most well-known data set in machine learning, the MNIST-Digits, or MNIST. An alternative representation is constructed, called MNIST-PD. This construction captures the topology of the digits using persistence diagrams, a product of persistent homology. We show that the models are more effective when trained on MNIST compared to MNIST-PD. Promising evidence reveals that the topology is learned by the algorithms. / Thesis / Master of Science (MSc)

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Modern data science applications demand solving large-scale optimization problems. The prevalent approaches are first-order methods, valued for their scalability. These methods are implemented to tackle highly irregular problems where assumptions of convexity and smoothness are untenable. Seeking to deepen the understanding of these methods, we study first-order methods with constant step size for minimizing locally Lipschitz tame functions. To do so, we propose notions of discrete Lyapunov stability for optimization methods. Concerning common first-order methods, we provide necessary and sufficient conditions for stability. We also show that certain local minima can be unstable, without additional noise in the method. Our analysis relies on the connection between the iterates of the first-order methods and continuous-time dynamics.

We develop the necessary theory in algebraic geometry to place Bayesian networks into the realm of algebraic statistics. This allows us to create an algebraic geometry--statistics dictionary. In particular, we study the algebraic varieties defined by the conditional independence statements of Bayesian networks. A complete algebraic classification, in terms of primary decomposition of polynomial ideals, is given for Bayesian networks on at most five random variables. Hidden variables are related to the geometry of higher secant varieties. Moreover, a complete algebraic classification, in terms of generating sets of polynomial ideals, is given for Bayesian networks on at most three random variables and one hidden variable. The relevance of these results for model selection is discussed. / Ph. D.