Mathematical Modeling and Computation – serie

Here is a clearly written introduction to three central areas of inverse problems: inverse problems in electromagnetic scattering theory, inverse spectral theory, and inverse problems in quantum scattering theory. Inverse problems, one of the most attractive parts of applied mathematics, attempt to obtain information about structures by nondestructive measurements. Based on a series of lectures presented by three of the authors, all experts in the field, the book provides a quick and easy way for readers to become familiar with the area through a survey of recent developments in inverse spectral and inverse scattering problems.In the opening chapter, Päivärinta collects the mathematical tools needed in the subsequent chapters and gives references for further study. Colton's chapter focuses on electromagnetic scattering problems. As an application he considers the problem of detecting and monitoring leukemia. Rundell's chapter deals with inverse spectral problems. He describes several exact and algorithmic methods for reconstructing an unknown function from the spectral data. Chadan provides an introduction to quantum mechanical inverse scattering problems. As an application he explains the celebrated method of Gardner, Greene, Kruskal, and Miura for solving nonlinear evolution equations such as the Korteweg_de Vries equation. Each chapter provides full references for further study.

Since the advent of computerized tomography in radiology, many imaging techniques have been introduced in medicine, science, and technology. This book describes the state of the art of the mathematical theory and numerical analysis of imaging. The authors survey and provide a unified view of imaging techniques, provide the necessary mathematical background and common framework, and give a detailed analysis of the numerical algorithms. This book not only reflects the theoretical progress and the growth of the field in the last 10 years but also serves as an excellent reference. It will provide readers with a superior understanding of the mathematical principles behind imaging and will enable them to write state-of-the-art software as a result.Mathematical Methods in Image Reconstruction provides a very detailed description of two-dimensional algorithms. For three-dimensional algorithms, the authors derive exact and approximate inversion formulas for specific imaging devices and describe their algorithmic implementation (which by and large parallels the two-dimensional algorithms). Integral geometry is surveyed as far as is necessary for imaging purposes; imaging techniques based on or related to integral geometry are briefly described in the section on tomography.Some of the applications covered in the book include computerized tomography, magnetic resonance imaging, emission tomography, electron microscopy, ultrasound transmission tomography, industrial tomography, seismic tomography, impedance tomography, and NIR imaging. The authors provide the necessary mathematical background and common mathematical framework needed to understand the book. Knowledge of tomography literature from the 1980s will be useful to the reader.

Mathematical modeling - the ability to apply mathematical concepts and techniques to real-life systems - has expanded considerably over the last decades, making it impossible to cover all of its aspects in one course or textbook. Continuum Modeling in the Physical Sciences provides an extensive exposition of the general principles and methods of this growing field with a focus on applications in the natural sciences. The authors present a thorough treatment of mathematical modeling from the elementary level to more advanced concepts.Most of the chapters are devoted to a discussion of central issues such as dimensional analysis, conservation principles, balance laws, constitutive relations, stability, robustness, and variational methods, and are accompanied by numerous real-life examples. Readers will benefit from the exercises placed throughout the text and the Challenging Problems sections found at the ends of several chapters. The last chapter is devoted to elaborated case studies in polymer dynamics, fiber spinning, water waves, and waveguide optics.

Modern financial mathematics relies on the theory of random processes in time, reflecting the erratic fluctuations in financial markets.This book introduces the fascinating area of financial mathematics and its calculus in an accessible manner geared toward undergraduate students. Using little high-level mathematics, the author presents the basic methods for evaluating financial options and building financial simulations.By emphasizing relevant applications and illustrating concepts with colour graphics, Elementary Calculus of Financial Mathematics presents the crucial concepts needed to understand financial options among these fluctuations. Among the topics covered are the binomial lattice model for evaluating financial options, the Black–Scholes and Fokker–Planck equations, and the interpretation of Ito’s formula in financial applications. Each chapter includes exercises for student practice and the appendices offer MATLAB® and SCILAB code as well as alternate proofs of the Fokker–Planck equation and Kolmogorov backward equation.

Presents cutting-edge developments in the theory and experiments of nonlinear waves. Its comprehensive coverage of analytical methods for nonintegrable systems is the first of its kind. It also covers in great depth analytical methods for integrable equations, and comprehensively describes efficient numerical methods for all major aspects of nonlinear wave computations. In addition, this book presents the latest experiments on nonlinear waves in optical systems and Bose–Einstein condensates, especially in periodic media. The book contains a large number of simple and efficient MATLAB® codes for various nonlinear wave computations, which readers can easily adapt to solve their own problems. The codes can also be found on an associated Web page.

Climate modeling and simulation teach us about past, present, and future conditions of life on earth and help us understand observations about the changing atmosphere and ocean and terrestrial ecology.Focusing on high-end modeling and simulation of earth's climate, Climate Modeling for Scientists and Engineers:Presents observations about the general circulations of the earth and the partial differential equations used to model the dynamics of weather and climate.Covers numerical methods for geophysical flows in more detail than many other texts.Discusses parallel algorithms and the role of high-performance computing used in the simulation of weather and climate.Provides supplemental lectures and MATLAB? exercises on an associated Web page.

Differential equations are the basis for models of any physical systems that exhibit smooth change. This book combines much of the material found in a traditional course on ordinary differential equations with an introduction to the more modern theory of dynamical systems. Applications of this theory to physics, biology, chemistry, and engineering are shown through examples in such areas as population modeling, fluid dynamics, electronics, and mechanics.Differential Dynamical Systems begins with coverage of linear systems, including matrix algebra; the focus then shifts to foundational material on nonlinear differential equations, making heavy use of the contraction-mapping theorem. Subsequent chapters deal specifically with dynamical systems concepts—flow, stability, invariant manifolds, the phase plane, bifurcation, chaos, and Hamiltonian dynamics. Revisions include simplified and clarified proofs of a number of theorems, an expanded introduction to function spaces, additional exercises, and the correction of typographical errors.Throughout the book, the author includes exercises to help students develop an analytical and geometrical understanding of dynamics. Many of the exercises and examples are based on applications and some involve computation; an appendix offers simple codes written in Maple®, Mathematica®, and MATLAB® software to give students practice with computation applied to dynamical systems problems.

Implicit-explicit (IMEX) time discretization methods have proven to be highly effective for the numerical solution of a wide class of evolutionary partial differential equations (PDEs) across various contexts. These methods have become mainstream for solving evolutionary PDEs, particularly in the fields of hyperbolic and kinetic equations.The first book on the subject, Implicit-Explicit Methods for Evolutionary Partial Differential Equations provides an in-depth yet accessible approach. The authors summarize and illustrate the construction, analysis, and application of IMEX methods using examples, test cases, and implementation details; guide readers through the various methods and teach them how to select and use the one most appropriate for their needs; and demonstrate how to identify stiff terms and effectively implement high-order methods in time for a variety of systems of PDEs.Readers interested in learning modern techniques for the effective numerical solution of evolutionary PDEs with multiple time scales will find in this book a unified, compact, and accessible treatment.

Wavelets continue to be powerful mathematical tools that can be used to solve problems for which the Fourier (spectral) method does not perform well or cannot handle. This book is for engineers, applied mathematicians, and other scientists who want to learn about using wavelets to analyze, process, and synthesize images and signals. Applications are described in detail and there are step-by-step instructions about how to construct and apply wavelets. The only mathematically rigorous monograph written by a mathematician specifically for nonspecialists, it describes the basic concepts of these mathematical techniques, outlines the procedures for using them, compares the performance of various approaches, and provides information for problem solving, putting the reader at the forefront of current research.New methods and important topics described in the book include the following:Multidimensional and multifrequency wavelet transforms (including wavelet packets). This approach gives the user more flexibility in applying wavelets to study multidimensional data sets.Spline wavelets with uniform or arbitrary knots on a bounded interval (with methods to construct these wavelets, their duals, as well as all decomposition and reconstruction matrices). This tool allows analysis and synthesis of discrete data on uniform or nonuniform sample sites without any boundary effect.Wavelets as a mathematical tool for waveform matching, signal segmentation, and time-frequency localization as well as effective implementation and fast computation.Procedures to construct all the well-known wavelets and to find their corresponding filter sequences.Detailed comparisons of the most popular wavelets and tables of values for evaluating their filtering performance.

Here is an overview of modern computational stabilization methods for linear inversion, with applications to a variety of problems in audio processing, medical imaging, tomography, seismology, astronomy, and other areas.Rank-deficient problems involve matrices that are either exactly or nearly rank deficient. Such problems often arise in connection with noise suppression and other problems where the goal is to suppress unwanted disturbances of the given measurements.Discrete ill-posed problems arise in connection with the numerical treatment of inverse problems, where one typically wants to compute information about some interior properties using exterior measurements. Examples of inverse problems are image restoration and tomography, where one needs to improve blurred images or reconstruct pictures from raw data.This book describes, in a common framework, new and existing numerical methods for the analysis and solution of rank-deficient and discrete ill-posed problems. The emphasis is on insight into the stabilizing properties of the algorithms and on the efficiency and reliability of the computations. The setting is that of numerical linear algebra rather than abstract functional analysis, and the theoretical development is complemented with numerical examples and figures that illustrate the features of the various algorithms.