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: 5730

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: 711

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: 6219

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: 3582

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: 1721

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: 4147

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.

Chebyshev and Fourier Spectral Methods

Second Revised Edition

Author: John P. Boyd

Publisher: Courier Corporation

ISBN: 0486141926

Category: Mathematics

Page: 688

View: 2917

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

View: 8067

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: 8850

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.

Certified Reduced Basis Methods for Parametrized Partial Differential Equations

Author: Jan S Hesthaven,Gianluigi Rozza,Benjamin Stamm

Publisher: Springer

ISBN: 3319224700

Category: Mathematics

Page: 131

View: 9902

This book provides a thorough introduction to the mathematical and algorithmic aspects of certified reduced basis methods for parametrized partial differential equations. Central aspects ranging from model construction, error estimation and computational efficiency to empirical interpolation methods are discussed in detail for coercive problems. More advanced aspects associated with time-dependent problems, non-compliant and non-coercive problems and applications with geometric variation are also discussed as examples.

Time Frequency Analysis

Author: Boualem Boashash

Publisher: Elsevier

ISBN: 9780080543055

Category: Technology & Engineering

Page: 770

View: 2572

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: 9917

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.

Dynamical Systems and Numerical Analysis

Author: Andrew Stuart,A. R. Humphries

Publisher: Cambridge University Press

ISBN: 9780521645638

Category: Mathematics

Page: 685

View: 2504

This book unites the study of dynamical systems and numerical solution of differential equations. The first three chapters contain the elements of the theory of dynamical systems and the numerical solution of initial-value problems. In the remaining chapters, numerical methods are formulted as dynamical systems and the convergence and stability properties of the methods are examined. Topics studied include the stability of numerical methods for contractive, dissipative, gradient and Hamiltonian systems together with the convergence properties of equilibria, periodic solutions and strage attractors under numerical approximation. This book will be an invaluable tool for graduate students and researchers in the fields of numerical analysis and dynamical systems.

Theory of Solidification

Author: Stephen H. Davis

Publisher: Cambridge University Press

ISBN: 9781139429634

Category: Science

Page: N.A

View: 1149

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: 9321

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

Finite Volume Methods for Hyperbolic Problems

Author: Randall J. LeVeque

Publisher: Cambridge University Press

ISBN: 1139434187

Category: Mathematics

Page: N.A

View: 7217

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: 4746

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.