In Computed Tomography: Algorithms, Insight, and Just Enough Theory, readers will learn about the fundamental computational methods used for image reconstruction in computed tomography (CT). Unique in its emphasis on the interplay of modeling, computing, and algorithm development, the book presents underlying mathematical models for motivating and deriving the basic principles of CT reconstruction methods, along with insight into their advantages, limitations, and computational aspects. Computed Tomography: Algorithms, Insight, and Just Enough Theory: Develops the mathematical and computational aspects of three main classes of reconstruction methods. Emphasizes the link between CT and numerical methods, which is rarely found in current literature. Describes the effects of incomplete data using both microlocal analysis and the singular value decomposition (SVD). Contains computer exercises using MATLAB that allow readers to experiment with the algorithms and make the book suitable for teaching and self-study. This book is aimed at students, researchers, and practitioners. As a textbook, it is appropriate for the following courses: Advanced Numerical Analysis, Special Topics on Numerical Analysis, Topics on Data Science, Topics on Numerical Optimization, and Topics on Approximation Theory.

Chordal graphs play a central role in techniques for exploiting sparsity in large semidefinite optimization problems, and in related convex optimization problems involving sparse positive semidefinite matrices. Chordal graph properties are also fundamental to several classical results in combinatorial optimization, linear algebra, statistics, signal processing, machine learning, and nonlinear optimization. This book covers the theory and applications of chordal graphs, with an emphasis on algorithms developed in the literature on sparse Cholesky factorization. These algorithms are formulated as recursions on elimination trees, supernodal elimination trees, or clique trees associated with the graph. The best known example is the multifrontal Cholesky factorization algorithm but similar algorithms can be formulated for a variety of related problems, such as the computation of the partial inverse of a sparse positive definite matrix, positive semidefinite and Euclidean distance matrix completion problems, and the evaluation of gradients and Hessians of logarithmic barriers for cones of sparse positive semidefinite matrices and their dual cones. This monograph shows how these techniques can be applied in algorithms for sparse semidefinite optimization. It also points out the connections with related topics outside semidefinite optimization, such as probabilistic networks, matrix completion problems, and partial separability in nonlinear optimization.