Multiscale Wavelet Methods For Partial Differential Equations
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Multiscale Wavelet Methods For Partial Differential Equations
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Author : Wolfgang Dahmen
language : en
Publisher: Elsevier
Release Date : 1997-08-13
Multiscale Wavelet Methods For Partial Differential Equations written by Wolfgang Dahmen and has been published by Elsevier this book supported file pdf, txt, epub, kindle and other format this book has been release on 1997-08-13 with Mathematics categories.
This latest volume in the Wavelets Analysis and Its Applications Series provides significant and up-to-date insights into recent developments in the field of wavelet constructions in connection with partial differential equations. Specialists in numerical applications and engineers in a variety of fields will find Multiscale Wavelet for Partial Differential Equations to be a valuable resource. - Covers important areas of computational mechanics such as elasticity and computational fluid dynamics - Includes a clear study of turbulence modeling - Contains recent research on multiresolution analyses with operator-adapted wavelet discretizations - Presents well-documented numerical experiments connected with the development of algorithms, useful in specific applications
Numerical Analysis Of Wavelet Methods
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Author : A. Cohen
language : en
Publisher: Elsevier
Release Date : 2003-04-29
Numerical Analysis Of Wavelet Methods written by A. Cohen and has been published by Elsevier this book supported file pdf, txt, epub, kindle and other format this book has been release on 2003-04-29 with Mathematics categories.
Since their introduction in the 1980's, wavelets have become a powerful tool in mathematical analysis, with applications such as image compression, statistical estimation and numerical simulation of partial differential equations. One of their main attractive features is the ability to accurately represent fairly general functions with a small number of adaptively chosen wavelet coefficients, as well as to characterize the smoothness of such functions from the numerical behaviour of these coefficients. The theoretical pillar that underlies such properties involves approximation theory and function spaces, and plays a pivotal role in the analysis of wavelet-based numerical methods. This book offers a self-contained treatment of wavelets, which includes this theoretical pillar and it applications to the numerical treatment of partial differential equations. Its key features are:1. Self-contained introduction to wavelet bases and related numerical algorithms, from the simplest examples to the most numerically useful general constructions.2. Full treatment of the theoretical foundations that are crucial for the analysisof wavelets and other related multiscale methods : function spaces, linear and nonlinear approximation, interpolation theory.3. Applications of these concepts to the numerical treatment of partial differential equations : multilevel preconditioning, sparse approximations of differential and integral operators, adaptive discretization strategies.
Wavelet Methods For Elliptic Partial Differential Equations
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Author : Karsten Urban
language : en
Publisher: OUP Oxford
Release Date : 2008-11-27
Wavelet Methods For Elliptic Partial Differential Equations written by Karsten Urban and has been published by OUP Oxford this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008-11-27 with Mathematics categories.
The origins of wavelets go back to the beginning of the last century and wavelet methods are by now a well-known tool in image processing (jpeg2000). These functions have, however, been used successfully in other areas, such as elliptic partial differential equations, which can be used to model many processes in science and engineering. This book, based on the author's course and accessible to those with basic knowledge of analysis and numerical mathematics, gives an introduction to wavelet methods in general and then describes their application for the numerical solution of elliptic partial differential equations. Recently developed adaptive methods are also covered and each scheme is complemented with numerical results, exercises, and corresponding software tools.
Adaptive Wavelet Frame Methods For Nonlinear Elliptic Problems
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Author : Jens Kappei
language : en
Publisher: Logos Verlag Berlin GmbH
Release Date : 2012-02-06
Adaptive Wavelet Frame Methods For Nonlinear Elliptic Problems written by Jens Kappei and has been published by Logos Verlag Berlin GmbH this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-02-06 with Mathematics categories.
Over the last ten years, adaptive wavelet methods have turned out to be a powerful tool in the numerical treatment of operator equations given on a bounded domain or closed manifold. In this work, we consider semi-nonlinear operator equations, including an elliptic linear operator as well as a nonlinear monotone one. Since the classical approach to construct a wavelet Riesz basis for the solution space is still afflicted with some notable problems, we use the weaker concept of wavelet frames to design an adaptive algorithm for the numerical solution of problems of this type. Choosing an appropriate overlapping decomposition of the given domain, a suitable frame system can be constructed easily. Applying it to the given continuous problem yields a discrete, bi-infinite nonlinear system of equations, which is shown to be solvable by a damped Richardson iteration method. We then successively introduce all building blocks for the numerical implementation of the iteration method. Here, we concentrate on the evaluation of the discrete nonlinearity, where we show that the previously developed auxiliary of tree-structured index sets can be generalized to the wavelet frame setting in a proper way. This allows an effective numerical treatment of the nonlinearity by so-called aggregated trees. Choosing the error tolerances appropriately, we show that our adaptive scheme is asymptotically optimal with respect to aggregated tree-structured index sets, i.e., it realizes the same convergence rate as the sequence of best N-term frame approximations of the solution respecting aggregated trees. Moreover, under the assumption of a sufficiently precise numerical quadrature method, the computational cost of our algorithm stays the same order as the number of wavelets used by it. The theoretical results are widely confirmed by one- and two-dimensional test problems over non-trivial bounded domains.
Wavelet Analysis And Multiresolution Methods
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Author : Tian-Xiao He
language : en
Publisher: CRC Press
Release Date : 2000-05-05
Wavelet Analysis And Multiresolution Methods written by Tian-Xiao He and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2000-05-05 with Mathematics categories.
This volume contains papers selected from the Wavelet Analysis and Multiresolution Methods Session of the AMS meeting held at the University of Illinois at Urbana-Champaign. The contributions cover: construction, analysis, computation and application of multiwavelets; scaling vectors; nonhomogenous refinement; mulivariate orthogonal and biorthogonal wavelets; and other related topics.
Partial Differential Equations
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Author : D. Sloan
language : en
Publisher: Elsevier
Release Date : 2012-12-02
Partial Differential Equations written by D. Sloan and has been published by Elsevier this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-02 with Mathematics categories.
/homepage/sac/cam/na2000/index.html7-Volume Set now available at special set price ! Over the second half of the 20th century the subject area loosely referred to as numerical analysis of partial differential equations (PDEs) has undergone unprecedented development. At its practical end, the vigorous growth and steady diversification of the field were stimulated by the demand for accurate and reliable tools for computational modelling in physical sciences and engineering, and by the rapid development of computer hardware and architecture. At the more theoretical end, the analytical insight into the underlying stability and accuracy properties of computational algorithms for PDEs was deepened by building upon recent progress in mathematical analysis and in the theory of PDEs. To embark on a comprehensive review of the field of numerical analysis of partial differential equations within a single volume of this journal would have been an impossible task. Indeed, the 16 contributions included here, by some of the foremost world authorities in the subject, represent only a small sample of the major developments. We hope that these articles will, nevertheless, provide the reader with a stimulating glimpse into this diverse, exciting and important field. The opening paper by Thomée reviews the history of numerical analysis of PDEs, starting with the 1928 paper by Courant, Friedrichs and Lewy on the solution of problems of mathematical physics by means of finite differences. This excellent survey takes the reader through the development of finite differences for elliptic problems from the 1930s, and the intense study of finite differences for general initial value problems during the 1950s and 1960s. The formulation of the concept of stability is explored in the Lax equivalence theorem and the Kreiss matrix lemmas. Reference is made to the introduction of the finite element method by structural engineers, and a description is given of the subsequent development and mathematical analysis of the finite element method with piecewise polynomial approximating functions. The penultimate section of Thomée's survey deals with `other classes of approximation methods', and this covers methods such as collocation methods, spectral methods, finite volume methods and boundary integral methods. The final section is devoted to numerical linear algebra for elliptic problems. The next three papers, by Bialecki and Fairweather, Hesthaven and Gottlieb and Dahmen, describe, respectively, spline collocation methods, spectral methods and wavelet methods. The work by Bialecki and Fairweather is a comprehensive overview of orthogonal spline collocation from its first appearance to the latest mathematical developments and applications. The emphasis throughout is on problems in two space dimensions. The paper by Hesthaven and Gottlieb presents a review of Fourier and Chebyshev pseudospectral methods for the solution of hyperbolic PDEs. Particular emphasis is placed on the treatment of boundaries, stability of time discretisations, treatment of non-smooth solutions and multidomain techniques. The paper gives a clear view of the advances that have been made over the last decade in solving hyperbolic problems by means of spectral methods, but it shows that many critical issues remain open. The paper by Dahmen reviews the recent rapid growth in the use of wavelet methods for PDEs. The author focuses on the use of adaptivity, where significant successes have recently been achieved. He describes the potential weaknesses of wavelet methods as well as the perceived strengths, thus giving a balanced view that should encourage the study of wavelet methods.
Wavelets In Numerical Simulation
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Author : Karsten Urban
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Wavelets In Numerical Simulation written by Karsten Urban and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
Sapere aude! Immanuel Kant (1724-1804) Numerical simulations playa key role in many areas of modern science and technology. They are necessary in particular when experiments for the underlying problem are too dangerous, too expensive or not even possible. The latter situation appears for example when relevant length scales are below the observation level. Moreover, numerical simulations are needed to control complex processes and systems. In all these cases the relevant problems may become highly complex. Hence the following issues are of vital importance for a numerical simulation: - Efficiency of the numerical solvers: Efficient and fast numerical schemes are the basis for a simulation of 'real world' problems. This becomes even more important for realtime problems where the runtime of the numerical simulation has to be of the order of the time span required by the simulated process. Without efficient solution methods the simulation of many problems is not feasible. 'Efficient' means here that the overall cost of the numerical scheme remains proportional to the degrees of freedom, i. e. , the numerical approximation is determined in linear time when the problem size grows e. g. to upgrade accuracy. Of course, as soon as the solution of large systems of equations is involved this requirement is very demanding.
Wavelets And Multiwavelets
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Author : Fritz Keinert
language : en
Publisher: CRC Press
Release Date : 2003-11-12
Wavelets And Multiwavelets written by Fritz Keinert and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2003-11-12 with Mathematics categories.
Theoretically, multiwavelets hold significant advantages over standard wavelets, particularly for solving more complicated problems, and hence are of great interest. Meeting the needs of engineers and mathematicians, this book provides a comprehensive overview of multiwavelets. The author presents the theory of wavelets from the viewpoint of genera
Fundamentals Of Wavelets
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Author : Jaideva C. Goswami
language : en
Publisher: John Wiley & Sons
Release Date : 2011-03-08
Fundamentals Of Wavelets written by Jaideva C. Goswami and has been published by John Wiley & Sons this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011-03-08 with Computers categories.
Most existing books on wavelets are either too mathematical or they focus on too narrow a specialty. This book provides a thorough treatment of the subject from an engineering point of view. It is a one-stop source of theory, algorithms, applications, and computer codes related to wavelets. This second edition has been updated by the addition of: a section on "Other Wavelets" that describes curvelets, ridgelets, lifting wavelets, etc a section on lifting algorithms Sections on Edge Detection and Geophysical Applications Section on Multiresolution Time Domain Method (MRTD) and on Inverse problems
Haar Wavelets
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Author : Ülo Lepik
language : en
Publisher: Springer Science & Business Media
Release Date : 2014-01-09
Haar Wavelets written by Ülo Lepik and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-01-09 with Technology & Engineering categories.
This is the first book to present a systematic review of applications of the Haar wavelet method for solving Calculus and Structural Mechanics problems. Haar wavelet-based solutions for a wide range of problems, such as various differential and integral equations, fractional equations, optimal control theory, buckling, bending and vibrations of elastic beams are considered. Numerical examples demonstrating the efficiency and accuracy of the Haar method are provided for all solutions.