Mathematical Modeling and Computation - Böcker

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 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.

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.

The field of mathematical biology is growing rapidly. Questions about infectious diseases, heart attacks, cell signaling, cell movement, ecology, environmental changes, and genomics are now being analyzed using mathematical and computational methods. A Course in Mathematical Biology teaches all aspects of modern mathematical modeling and is specifically designed to introduce undergraduate students to problem solving in the context of biology.Divided into three parts, the book covers basic analytical modeling techniques and model validation methods; introduces computational tools used in the modeling of biological problems; and provides a source of open-ended problems from epidemiology, ecology, and physiology. All chapters include realistic biological examples, and there are many exercises related to biological questions. In addition, the book includes 25 open-ended research projects that can be used by students. The book is accompanied by a Web site that contains solutions to most of the exercises and a tutorial for the implementation of the computational modeling techniques. Calculations can be done in modern computing languages such as Maple, Mathematica, and MATLAB®.

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.

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.

This book provides an overview of the myriad methods for applying dynamical systems techniques to PDEs and highlights the impact of PDE methods on dynamical systems. Also included are many nonlinear evolution equations, which have been benchmark models across the sciences, and examples and techniques to strengthen preparation for research. PDE Dynamics: An Introduction is intended for senior undergraduate students, beginning graduate students, and researchers in applied mathematics, theoretical physics, and adjacent disciplines. Structured as a textbook or seminar reference, it can be used in courses titled Dynamics of PDEs, PDEs 2, Dynamical Systems 2, Evolution Equations, or Infinite-Dimensional Dynamics.