Bücher der Reihe Cambridge Monographs on Applied and Computational Mathematics

This book presents the basic concepts from the emerging field of computational topology that combines topology theory with the power of computing to solve problems in diverse fields. Written from a computer science perspective, the book enables non-specialists to grasp the ideas and so participate in current research in computational topology.

Recovering the phase of the Fourier transform is a ubiquitous problem in imaging applications from astronomy to nanoscale X-ray diffraction imaging. Despite the efforts of a multitude of scientists, from astronomers to mathematicians, there is, as yet, no satisfactory theoretical or algorithmic solution to this class of problems. Written for mathematicians, physicists and engineers working in image analysis and reconstruction, this book introduces a conceptual, geometric framework for the analysis of these problems, leading to a deeper understanding of the essential, algorithmically independent, difficulty of their solutions. Using this framework, the book studies standard algorithms and a range of theoretical issues in phase retrieval and provides several new algorithms and approaches to this problem with the potential to improve the reconstructed images. The book is lavishly illustrated with the results of numerous numerical experiments that motivate the theoretical development and place it in the context of practical applications.

This book offers a comprehensive introduction to the theory of linear and nonlinear Volterra integral equations (VIEs), ranging from Volterra's fundamental contributions and the resulting classical theory to more recent developments that include Volterra functional integral equations with various kinds of delays, VIEs with highly oscillatory kernels, and VIEs with non-compact operators. It will act as a 'stepping stone' to the literature on the advanced theory of VIEs, bringing the reader to the current state of the art in the theory. Each chapter contains a large number of exercises, extending from routine problems illustrating or complementing the theory to challenging open research problems. The increasingly important role of VIEs in the mathematical modelling of phenomena where memory effects play a key role is illustrated with some 30 concrete examples, and the notes at the end of each chapter feature complementary references as a guide to further reading.

Geometric integrators are time-stepping methods, designed such that they exactly satisfy conservation laws, symmetries or symplectic properties of a system of differential equations. In this book the authors outline the principles of geometric integration and demonstrate how they can be applied to provide efficient numerical methods for simulating conservative models. Beginning from basic principles and continuing with discussions regarding the advantageous properties of such schemes, the book introduces methods for the N-body problem, systems with holonomic constraints, and rigid bodies. More advanced topics treated include high-order and variable stepsize methods, schemes for treating problems involving multiple time-scales, and applications to molecular dynamics and partial differential equations. The emphasis is on providing a unified theoretical framework as well as a practical guide for users. The inclusion of examples, background material and exercises enhance the usefulness of the book for self-instruction or as a text for a graduate course on the subject.

This book is the first to give an accessible overview of the Christoffel-Darboux kernel as a novel, powerful, yet simple, tool in statistical data analysis. It offers non-expert, graduate-level readers a rapid and informative introduction to an inter-disciplinary subject, with numerous ramifications and intriguing open questions.

Structured population models can be used to describe processes such as infection transmission, cell growth and traffic flows. This book introduces the mathematical underpinnings of these applications, providing a unified framework for the study (numerical and analytic) of transport equations, and collecting results from various fields in one place.

Starting from classical linear approximation, this is a self-contained presentation of modern multivariate approximation theory that explores its connections with other areas of mathematics. The prerequisites are no more than standard undergraduate mathematics, so the book will be accessible to graduate students and non-specialists.

The recent appearance of wavelets as a new computational tool in applied mathematics has given a new impetus to the field of numerical analysis of Fredholm integral equations. This book gives an account of the state of the art in the study of fast multiscale methods for solving these equations based on wavelets. The authors begin by introducing essential concepts and describing conventional numerical methods. They then develop fast algorithms and apply these to solving linear, nonlinear Fredholm integral equations of the second kind, ill-posed integral equations of the first kind and eigen-problems of compact integral operators. Theorems of functional analysis used throughout the book are summarised in the appendix. The book is an essential reference for practitioners wishing to use the new techniques. It may also be used as a text, with the first five chapters forming the basis of a one-semester course for advanced undergraduates or beginning graduates.

Many texts teach solution methods for differential equations, but this is the first that explains how to extend these methods to difference equations. It assumes no prior knowledge of difference equations, making it ideal for newcomers, but also contains much new material that will interest researchers in the field.

Vladimir Temlyakov guides the reader through the fundamental results and introduces two other hot topics in numerical mathematics: learning theory and compressed sensing. Researchers will welcome this first book on the subject. It is ideal for graduate courses and includes many important open problems that provide ideas for further research.

Modern Computer Arithmetic focuses on arbitrary-precision algorithms for efficiently performing arithmetic operations such as addition, multiplication and division, and their connections to topics such as modular arithmetic, greatest common divisors, the Fast Fourier Transform (FFT), and the computation of elementary and special functions. Brent and Zimmermann present algorithms that are ready to implement in your favourite language, while keeping a high-level description and avoiding too low-level or machine-dependent details. The book is intended for anyone interested in the design and implementation of efficient high-precision algorithms for computer arithmetic, and more generally efficient multiple-precision numerical algorithms. It may also be used in a graduate course in mathematics or computer science, for which exercises are included. These vary considerably in difficulty, from easy to small research projects, and expand on topics discussed in the text. Solutions to selected exercises are available from the authors.

Computational simulation of scientific phenomena and engineering problems often depends on solving linear systems with a large number of unknowns. This book gives insight into the construction of iterative methods for the solution of such systems and helps the reader to select the best solver for a given class of problems. The emphasis is on the main ideas and how they have led to efficient solvers such as CG, GMRES, and BI-CGSTAB. The author also explains the main concepts behind the construction of preconditioners. The reader is encouraged to gain experience by analysing numerous examples that illustrate how best to exploit the methods. The book also hints at many open problems and as such it will appeal to established researchers. There are many exercises that motivate the material and help students to understand the essential steps in the analysis and construction of algorithms.

This book explains and develops new applications of continuous symmetry groups (Lie groups), in particular, the symbolic manipulation of invariants of group actions. The language used is primarily that of undergraduate calculus. More sophisticated ideas from differential topology and Lie theory are explained from scratch using worked examples and exercises.

This book explains how, when and why the pseudospectral approach works.

Linked Twist Maps can provide a unifying framework for understanding many types of fluid mixing, ranging from the very small to the very large, from fluids to solids. The authors discuss the definition and construction of LTMs, provide examples of specific mixers, and present a number of open problems.

Sure to be influential, this book lays the foundations for the use of algebraic geometry in statistical learning theory. Many widely used statistical models are singular: mixture models, neural networks, HMMs, and Bayesian networks are major examples. The theory achieved here underpins accurate estimation techniques in the presence of singularities.

The goal of learning theory is to approximate a function from sample values. This is a general overview of the theoretical foundations, and is the first book to emphasize the approximation theory viewpoint. This emphasis provides a balanced approach, and will attract mathematicians to the problems raised.

After developing the basics of a sampling theory and its connections to various geometric and topological properties, the author describes a suite of algorithms that have been designed for the reconstruction problem, including algorithms for surface reconstruction from dense samples, from samples that are not adequately dense and from noisy samples.

Spectral methods are well-suited to solve problems modeled by time-dependent partial differential equations: they are fast, efficient and accurate and widely used by mathematicians and practitioners. This class-tested 2007 introduction, the first on the subject, is ideal for graduate courses, or self-study. The authors describe the basic theory of spectral methods, allowing the reader to understand the techniques through numerous examples as well as more rigorous developments. They provide a detailed treatment of methods based on Fourier expansions and orthogonal polynomials (including discussions of stability, boundary conditions, filtering, and the extension from the linear to the nonlinear situation). Computational solution techniques for integration in time are dealt with by Runge-Kutta type methods. Several chapters are devoted to material not previously covered in book form, including stability theory for polynomial methods, techniques for problems with discontinuous solutions, round-off errors and the formulation of spectral methods on general grids. These will be especially helpful for practitioners.

Mesh generation combines different approaches to problem solving from mathematics, computer science, and engineering. This book emphasizes topics that are elementary, attractive, useful, interesting, and lend themselves to teaching, making it an ideal graduate text for courses on mesh generation.

A comprehensive bibliography rounds off what will prove a very valuable work.

A complete self-contained introduction to the theory of scattered data approximation. Written with graduates and researchers in mind, the text brings together much of the necessary background material into a single treatment and provides students with complete proofs to the theory developed within.

This monograph presents the GRP algorithm and is accessible to researchers and graduate students alike.

High-order numerical methods provide an efficient approach to simulating many physical problems. This book considers the range of mathematical, engineering, and computer science topics that form the foundation of high-order numerical methods for the simulation of incompressible fluid flows in complex domains. Introductory chapters present high-order spatial and temporal discretizations for one-dimensional problems. These are extended to multiple space dimensions with a detailed discussion of tensor-product forms, multi-domain methods, and preconditioners for iterative solution techniques. Numerous discretizations of the steady and unsteady Stokes and Navier-Stokes equations are presented, with particular attention given to enforcement of incompressibility. Advanced discretizations, implementation issues, and parallel and vector performance are considered in the closing sections. Numerous examples are provided throughout to illustrate the capabilities of high-order methods in actual applications. Computer scientists, engineers and applied mathematicians interested in developing software for solving flow problems will find this book a valuable reference.

This 1996 book unites the study of dynamical systems and numerical solution of differential equations. It will be an invaluable tool for graduate students and researchers in the fields of numerical analysis and dynamical systems.

This book provides an extensive introduction to the numerical solution of a large class of integral equations. Each chapter concludes with a discussion of the literature and a large bibliography serves as an extended resource for students and researchers needing more information on solving particular integral equations.

This book generalises the classical theory of orthogonal polynomials on the complex unit circle or on the real line to orthogonal rational functions whose poles are among a prescribed set of complex numbers. This theory has applications in theoretical real analysis, complex analysis, approximation theory, numerical analysis, and electrical engineering.

This book is an introduction to level set methods and fast marching methods, which are powerful numerical techniques for analyzing and computing interface motion in a host of settings. It will be a useful resource for mathematicians, applied scientists, practising engineers and computer graphic artists.