Spectral Methods for Time-Dependent Problems


Author: Jan S. Hesthaven,Sigal Gottlieb,David Gottlieb
Publisher: Cambridge University Press
ISBN: 113945952X
Category: Mathematics
Page: N.A
View: 884

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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.

Spectral Methods for Time-Dependent Problems


Author: Jan S. Hesthaven,Sigal Gottlieb,David Gottlieb
Publisher: Cambridge University Press
ISBN: 113945952X
Category: Mathematics
Page: N.A
View: 3401

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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.

Lectures on the Theory of Water Waves


Author: Thomas J. Bridges,Mark D. Groves,David P. Nicholls
Publisher: Cambridge University Press
ISBN: 1316558940
Category: Science
Page: N.A
View: 8524

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In the summer of 2014 leading experts in the theory of water waves gathered at the Newton Institute for Mathematical Sciences in Cambridge for four weeks of research interaction. A cross-section of those experts was invited to give introductory-level talks on active topics. This book is a compilation of those talks and illustrates the diversity, intensity, and progress of current research in this area. The key themes that emerge are numerical methods for analysis, stability and simulation of water waves, transform methods, rigorous analysis of model equations, three-dimensionality of water waves, variational principles, shallow water hydrodynamics, the role of deterministic and random bottom topography, and modulation equations. This book is an ideal introduction for PhD students and researchers looking for a research project. It may also be used as a supplementary text for advanced courses in mathematics or fluid dynamics.

Partial Differential Equations: Modeling, Analysis and Numerical Approximation


Author: Hervé Le Dret,Brigitte Lucquin
Publisher: Birkhäuser
ISBN: 3319270672
Category: Mathematics
Page: 395
View: 2383

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This book is devoted to the study of partial differential equation problems both from the theoretical and numerical points of view. After presenting modeling aspects, it develops the theoretical analysis of partial differential equation problems for the three main classes of partial differential equations: elliptic, parabolic and hyperbolic. Several numerical approximation methods adapted to each of these examples are analyzed: finite difference, finite element and finite volumes methods, and they are illustrated using numerical simulation results. Although parts of the book are accessible to Bachelor students in mathematics or engineering, it is primarily aimed at Masters students in applied mathematics or computational engineering. The emphasis is on mathematical detail and rigor for the analysis of both continuous and discrete problems.

A Practical Guide to Spectral Computational Methods


Author: George Rawitscher,Victo dos Santos Filho,Thiago C. Peixoto,Lauro Tomio
Publisher: Springer
ISBN: 3319427032
Category: Science
Page: 194
View: 1376

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This book disseminates basic aspects of modern spectral computational methods, which are not generally taught in traditional courses. The main advantage of these methods is that they take into account all available information at the same time, rather only the information available at a limited number of meshpoints. That leads to more complicated matrix equations, but the elegance, speed, and accuracy of the method more than compensates for this drawback. The method consists in expanding the function to be calculated into a set of appropriate basis functions (generally orthogonal polynomials), and the respective expansion coefficients are obtained via collocation equations. In the various chapters, the authors examine the usually rapid convergence of the spectral expansions and the improved accuracy that results when nonequispaced support points are used, in contrast to the equispaced points used in finite difference methods, and, in particular, they demonstrate the enhanced accuracy obtained in the solution of integral equations. It includes an informative introduction to the old and new computational methods with numerous practical examples, while at the same time raising awareness of the errors that each of the available algorithms introduces into the specific solution. It is a valuable resource for graduate students wishing to compare the available computational methods (canned or not canned) and judge which is the most suitable to solve the particular scientific problem they are confronting.

Algebraic Geometry and Statistical Learning Theory


Author: Sumio Watanabe
Publisher: Cambridge University Press
ISBN: 0521864674
Category: Computers
Page: 286
View: 7329

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Sure to be influential, Watanabe's book lays the foundations for the use of algebraic geometry in statistical learning theory. Many models/machines are singular: mixture models, neural networks, HMMs, Bayesian networks, stochastic context-free grammars are major examples. The theory achieved here underpins accurate estimation techniques in the presence of singularities.

Nodal Discontinuous Galerkin Methods

Algorithms, Analysis, and Applications
Author: Jan S. Hesthaven,Tim Warburton
Publisher: Springer Science & Business Media
ISBN: 0387720650
Category: Mathematics
Page: 502
View: 4130

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This book offers an introduction to the key ideas, basic analysis, and efficient implementation of discontinuous Galerkin finite element methods (DG-FEM) for the solution of partial differential equations. It covers all key theoretical results, including an overview of relevant results from approximation theory, convergence theory for numerical PDE’s, and orthogonal polynomials. Through embedded Matlab codes, coverage discusses and implements the algorithms for a number of classic systems of PDE’s: Maxwell’s equations, Euler equations, incompressible Navier-Stokes equations, and Poisson- and Helmholtz equations.

Chebyshev and Fourier Spectral Methods

Second Revised Edition
Author: John P. Boyd
Publisher: Courier Corporation
ISBN: 0486141926
Category: Mathematics
Page: 688
View: 3597

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Completely revised text applies spectral methods to boundary value, eigenvalue, and time-dependent problems, but also covers cardinal functions, matrix-solving methods, coordinate transformations, much more. Includes 7 appendices and over 160 text figures.

Learning Theory

An Approximation Theory Viewpoint
Author: Felipe Cucker,Ding Xuan Zhou
Publisher: Cambridge University Press
ISBN: 1139462865
Category: Computers
Page: N.A
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The goal of learning theory is to approximate a function from sample values. To attain this goal learning theory draws on a variety of diverse subjects, specifically statistics, approximation theory, and algorithmics. Ideas from all these areas blended to form a subject whose many successful applications have triggered a rapid growth during the last two decades. This is the first book to give a general overview of the theoretical foundations of the subject emphasizing the approximation theory, while still giving a balanced overview. It is based on courses taught by the authors, and is reasonably self-contained so will appeal to a broad spectrum of researchers in learning theory and adjacent fields. It will also serve as an introduction for graduate students and others entering the field, who wish to see how the problems raised in learning theory relate to other disciplines.

Simulating Hamiltonian Dynamics


Author: Benedict Leimkuhler,Sebastian Reich
Publisher: Cambridge University Press
ISBN: 9780521772907
Category: Mathematics
Page: 379
View: 8776

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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.

Time Frequency Analysis


Author: Boualem Boashash
Publisher: Elsevier
ISBN: 9780080543055
Category: Technology & Engineering
Page: 770
View: 2109

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Time Frequency Signal Analysis and Processing covers fundamental concepts, principles and techniques, treatment of specialised and advanced topics, methods and applications, including results of recent research. This book deals with the modern methodologies, key techniques and concepts that form the core of new technologies used in IT, multimedia, telecommunications as well as most fields of engineering, science and technology. It focuses on advanced techniques and methods that allow a refined extraction and processing of information, allowing efficient and effective decision making that would not be possible with classical techniques. The Author, fellow of IEEE for Pioneering contributions to time-frequency analysis and signal processing education, is an expert in the field, having written over 300 papers on the subject over a period pf 25 years. This is a REAL book, not a mere collection of specialised papers, making it essential reading for researchers and practitioners in the field of signal processing. *The most comprehensive text and reference book published on the subject, all the most up to date research on this subject in one place *Key computer procedures and code are provided to assist the reader with practical implementations and applications *This book brings together the main knowledge of time-frequency signal analysis and processing, (TFSAP), from theory and applications, in a user-friendly reference suitable for both experts and beginners

Collocation Methods for Volterra Integral and Related Functional Differential Equations


Author: Hermann Brunner
Publisher: Cambridge University Press
ISBN: 9780521806152
Category: Mathematics
Page: 597
View: 2580

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Collocation based on piecewise polynomial approximation represents a powerful class of methods for the numerical solution of initial-value problems for functional differential and integral equations arising in a wide spectrum of applications, including biological and physical phenomena. The present book introduces the reader to the general principles underlying these methods and then describes in detail their convergence properties when applied to ordinary differential equations, functional equations with (Volterra type) memory terms, delay equations, and differential-algebraic and integral-algebraic equations. Each chapter starts with a self-contained introduction to the relevant theory of the class of equations under consideration. Numerous exercises and examples are supplied, along with extensive historical and bibliographical notes utilising the vast annotated reference list of over 1300 items. In sum, Hermann Brunner has written a treatise that can serve as an introduction for students, a guide for users, and a comprehensive resource for experts.

Theory of Solidification


Author: Stephen H. Davis
Publisher: Cambridge University Press
ISBN: 9781139429634
Category: Science
Page: N.A
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The processes of freezing and melting were present at the beginnings of the Earth and continue to dominate the natural and industrial worlds. The solidification of a liquid or the melting of a solid involves a complex interplay of many physical effects. This 2001 book presents in a systematic way the field of continuum solidification theory based on instability phenomena. An understanding of the physics is developed by using examples of increasing complexity with the object of creating a deep physical insight applicable to more complex problems. Applied mathematicians, engineers, physicists, and materials scientists will all find this volume of interest.

Data-Driven Computational Methods

Parameter and Operator Estimations
Author: John Harlim
Publisher: Cambridge University Press
ISBN: 1108472478
Category: Computers
Page: 169
View: 6897

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Describes computational methods for parametric and nonparametric modeling of stochastic dynamics. Aimed at graduate students, and suitable for self-study.

High-Order Methods for Incompressible Fluid Flow


Author: M. O. Deville,P. F. Fischer,E. H. Mund
Publisher: Cambridge University Press
ISBN: 9780521453097
Category: Mathematics
Page: 499
View: 1292

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This book covers the development of high-order numerical methods for the simulation of incompressible fluid flows in complex domains.

Finite Volume Methods for Hyperbolic Problems


Author: Randall J. LeVeque
Publisher: Cambridge University Press
ISBN: 1139434187
Category: Mathematics
Page: N.A
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This book, first published in 2002, contains an introduction to hyperbolic partial differential equations and a powerful class of numerical methods for approximating their solution, including both linear problems and nonlinear conservation laws. These equations describe a wide range of wave propagation and transport phenomena arising in nearly every scientific and engineering discipline. Several applications are described in a self-contained manner, along with much of the mathematical theory of hyperbolic problems. High-resolution versions of Godunov's method are developed, in which Riemann problems are solved to determine the local wave structure and limiters are then applied to eliminate numerical oscillations. These methods were originally designed to capture shock waves accurately, but are also useful tools for studying linear wave-propagation problems, particularly in heterogenous material. The methods studied are implemented in the CLAWPACK software package and source code for all the examples presented can be found on the web, along with animations of many of the simulations. This provides an excellent learning environment for understanding wave propagation phenomena and finite volume methods.

Practical Extrapolation Methods

Theory and Applications
Author: Avram Sidi
Publisher: Cambridge University Press
ISBN: 9780521661591
Category: Computers
Page: 519
View: 6463

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This book is concerned with the coherent treatment, including the derivation, analysis, and applications, of the most useful scalar extrapolation methods. Its importance is rooted in the fact that the methods it discusses are geared towards problems that arise commonly in scientific and engineering disciplines. It differs from existing books on the subject in that it concentrates on the most powerful nonlinear methods, presents in-depth treatments of them, and shows which methods are most effective for different classes of practical nontrivial problems, and also shows how to apply these methods to obtain best results.

Computer Vision

Algorithms and Applications
Author: Richard Szeliski
Publisher: Springer Science & Business Media
ISBN: 9781848829350
Category: Computers
Page: 812
View: 4938

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Computer Vision: Algorithms and Applications explores the variety of techniques commonly used to analyze and interpret images. It also describes challenging real-world applications where vision is being successfully used, both for specialized applications such as medical imaging, and for fun, consumer-level tasks such as image editing and stitching, which students can apply to their own personal photos and videos. More than just a source of “recipes,” this exceptionally authoritative and comprehensive textbook/reference also takes a scientific approach to basic vision problems, formulating physical models of the imaging process before inverting them to produce descriptions of a scene. These problems are also analyzed using statistical models and solved using rigorous engineering techniques. Topics and features: structured to support active curricula and project-oriented courses, with tips in the Introduction for using the book in a variety of customized courses; presents exercises at the end of each chapter with a heavy emphasis on testing algorithms and containing numerous suggestions for small mid-term projects; provides additional material and more detailed mathematical topics in the Appendices, which cover linear algebra, numerical techniques, and Bayesian estimation theory; suggests additional reading at the end of each chapter, including the latest research in each sub-field, in addition to a full Bibliography at the end of the book; supplies supplementary course material for students at the associated website, http://szeliski.org/Book/. Suitable for an upper-level undergraduate or graduate-level course in computer science or engineering, this textbook focuses on basic techniques that work under real-world conditions and encourages students to push their creative boundaries. Its design and exposition also make it eminently suitable as a unique reference to the fundamental techniques and current research literature in computer vision.

Difference Equations by Differential Equation Methods


Author: Peter E. Hydon
Publisher: Cambridge University Press
ISBN: 1139991701
Category: Mathematics
Page: N.A
View: 3337

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Most well-known solution techniques for differential equations exploit symmetry in some form. Systematic methods have been developed for finding and using symmetries, first integrals and conservation laws of a given differential equation. Here the author explains how to extend these powerful methods to difference equations, greatly increasing the range of solvable problems. Beginning with an introduction to elementary solution methods, the book gives readers a clear explanation of exact techniques for ordinary and partial difference equations. The informal presentation is suitable for anyone who is familiar with standard differential equation methods. No prior knowledge of difference equations or symmetry is assumed. The author uses worked examples to help readers grasp new concepts easily. There are 120 exercises of varying difficulty and suggestions for further reading. The book goes to the cutting edge of research; its many new ideas and methods make it a valuable reference for researchers in the field.