This book takes readers on a multi-perspective tour through state-of-the-art mathematical developments related to the numerical treatment of PDEs based on splines, and in particular isogeometric methods. A wide variety of research topics are covered, ranging from approximation theory to structured numerical linear algebra. More precisely, the book provides (i) a self-contained introduction to B-splines, with special focus on approximation and hierarchical refinement, (ii) a broad survey of numerical schemes for control problems based on B-splines and B-spline-type wavelets, (iii) an exhaustive description of methods for computing and analyzing the spectral distribution of discretization matrices, and (iv) a detailed overview of the mathematical and implementational aspects of isogeometric analysis. The text is the outcome of a C.I.M.E. summer school held in Cetraro (Italy), July 2017, featuring four prominent lecturers with different theoretical and application perspectives. The book may serve both as a reference and an entry point into further research.
To put the world of linear algebra to advanced use, it is not enough to merely understand the theory; there is a significant gap between the theory of linear algebra and its myriad expressions in nearly every computational domain. To bridge this gap, it is essential to process the theory by solving many exercises, thus obtaining a firmer grasp of its diverse applications. Similarly, from a theoretical perspective, diving into the literature on advanced linear algebra often reveals more and more topics that are deferred to exercises instead of being treated in the main text. As exercises grow more complex and numerous, it becomes increasingly important to provide supporting material and guidelines on how to solve them, supporting students’ learning process. This book provides precisely this type of supporting material for the textbook “Numerical Linear Algebra and Matrix Factorizations,” published as Vol. 22 of Springer’s Texts in Computational Science and Engineering series. Instead of omitting details or merely providing rough outlines, this book offers detailed proofs, and connects the solutions to the corresponding results in the textbook. For the algorithmic exercises the utmost level of detail is provided in the form of MATLAB implementations. Both the textbook and solutions are self-contained. This book and the textbook are of similar length, demonstrating that solutions should not be considered a minor aspect when learning at advanced levels.
After reading this book, students should be able to analyze computational problems in linear algebra such as linear systems, least squares- and eigenvalue problems, and to develop their own algorithms for solving them. Since these problems can be large and difficult to handle, much can be gained by understanding and taking advantage of special structures. This in turn requires a good grasp of basic numerical linear algebra and matrix factorizations. Factoring a matrix into a product of simpler matrices is a crucial tool in numerical linear algebra, because it allows us to tackle complex problems by solving a sequence of easier ones. The main characteristics of this book are as follows: It is self-contained, only assuming that readers have completed first-year calculus and an introductory course on linear algebra, and that they have some experience with solving mathematical problems on a computer. The book provides detailed proofs of virtually all results. Further, its respective parts can be used independently, making it suitable for self-study. The book consists of 15 chapters, divided into five thematically oriented parts. The chapters are designed for a one-week-per-chapter, one-semester course. To facilitate self-study, an introductory chapter includes a brief review of linear algebra.
Designed to provide tools for independent study, this book contains student-tested mathematical exercises joined with MATLAB programming exercises. Most chapters open with a review followed by theoretical and programming exercises, with detailed solutions provided for all problems including programs. Many of the MATLAB exercises are presented as Russian dolls: each question improves and completes the previous program and results are provided to validate the intermediate programs. The book offers useful MATLAB commands, advice on tables, vectors, matrices and basic commands for plotting. It contains material on eigenvalues and eigenvectors and important norms of vectors and matrices including perturbation theory; iterative methods for solving nonlinear and linear equations; polynomial and piecewise polynomial interpolation; Bézier curves; approximations of functions and integrals and more. The last two chapters considers ordinary differential equations including two point boundary value problems, and deal with finite difference methods for some partial differential equations. The format is designed to assist students working alone, with concise Review paragraphs, Math Hint footnotes on the mathematical aspects of a problem and MATLAB Hint footnotes with tips on programming.
An analysis of the interactions between pelagic food web processes and element cycling in lakes. While some findings are examined in terms of classical concepts from the ecological theory of predator-prey systems, special emphasis is placed on exploring how stoichiometric relationships between primary producers and herbivores influence the stability and persistence of planktonic food webs. The author develops simple dynamic models of the cycling of mineral nutrients through plankton algae and grazers, and then goes on to explore them both analytically and numerically. The results thus obtained are of great interest to both theoretical and experimental ecologists. Moreover, the models themselves are of immense practical use in the area of lake management.
On August 11, 1979, after a week of extraordinary monsoon rains in the Indian state of Gujarat, the two mile-long Machhu Dam-II disintegrated. The waters released from the dam’s massive reservoir rushed through the heavily populated downstream area, devastating the industrial city of Morbi and its surrounding agricultural villages. As the torrent’s thirty-foot-tall leading edge cut its way through the Machhu River valley, massive bridges gave way, factories crumbled, and thousands of houses collapsed. While no firm figure has ever been set on the disaster’s final death count, estimates in the flood’s wake ran as high as 25,000. Despite the enormous scale of the devastation, few people today have ever heard of this terrible event. This book tells, for the first time, the suspenseful and multifaceted story of the Machhu dam disaster. Based on over 130 interviews and extensive archival research, the authors recount the disaster and its aftermath in vivid firsthand detail. The book presents important findings culled from formerly classified government documents that reveal the long-hidden failures that culminated in one of the deadliest floods in history. The authors follow characters whose lives were interrupted and forever altered by the flood; provide vivid first-hand descriptions of the disaster and its aftermath; and shed light on the never-completed judicial investigation into the dam’s collapse.
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