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Wavelet Methods For Elliptic Partial Differential Equations
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Book Synopsis Wavelet Methods for Elliptic Partial Differential Equations by : Karsten Urban
Download or read book Wavelet Methods for Elliptic Partial Differential Equations written by Karsten Urban and published by OUP Oxford. This book was released on 2008-11-27 with total page 512 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Adaptive Wavelet Methods for Elliptic Partial Differential Equations with Random Operators by : Claude Jeffrey Gittelson
Download or read book Adaptive Wavelet Methods for Elliptic Partial Differential Equations with Random Operators written by Claude Jeffrey Gittelson and published by . This book was released on 2011 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Multiscale Wavelet Methods for Partial Differential Equations by : Wolfgang Dahmen
Download or read book Multiscale Wavelet Methods for Partial Differential Equations written by Wolfgang Dahmen and published by Elsevier. This book was released on 1997-08-13 with total page 587 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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
Book Synopsis Adaptive Wavelet Methods for Variational Formulations of Nonlinear Elliptic PDEs on Tensor-Product Domains by : Roland Pabel
Download or read book Adaptive Wavelet Methods for Variational Formulations of Nonlinear Elliptic PDEs on Tensor-Product Domains written by Roland Pabel and published by Logos Verlag Berlin GmbH. This book was released on 2015-09-30 with total page 336 pages. Available in PDF, EPUB and Kindle. Book excerpt: This thesis is concerned with the numerical solution of boundary value problems (BVPs) governed by nonlinear elliptic partial differential equations (PDEs). To iteratively solve such BVPs, it is of primal importance to develop efficient schemes that guarantee convergence of the numerically approximated PDE solutions towards the exact solution. The new adaptive wavelet theory guarantees convergence of adaptive schemes with fixed approximation rates. Furthermore, optimal, i.e., linear, complexity estimates of such adaptive solution methods have been established. These achievements are possible since wavelets allow for a completely new perspective to attack BVPs: namely, to represent PDEs in their original infinite dimensional realm. Wavelets in this context represent function bases with special analytical properties, e.g., the wavelets considered herein are piecewise polynomials, have compact support and norm equivalences between certain function spaces and the $ell_2$ sequence spaces of expansion coefficients exist. This theoretical framework is implemented in the course of this thesis in a truly dimensionally unrestricted adaptive wavelet program code, which allows one to harness the proven theoretical results for the first time when numerically solving the above mentioned BVPs. Numerical studies of 2D and 3D PDEs and BVPs demonstrate the feasibility and performance of the developed schemes. The BVPs are solved using an adaptive Uzawa algorithm, which requires repeated solution of nonlinear PDE sub-problems. This thesis presents for the first time a numerically competitive implementation of a new theoretical paradigm to solve nonlinear elliptic PDEs in arbitrary space dimensions with a complete convergence and complexity theory.
Book Synopsis Wavelet Methods for Elliptic Partial Differential Equations by : Karsten Urban
Download or read book Wavelet Methods for Elliptic Partial Differential Equations written by Karsten Urban and published by Numerical Mathematics and Scie. This book was released on 2009 with total page 509 pages. Available in PDF, EPUB and Kindle. Book excerpt: Wavelet methods are by now a well-known tool in image processing (jpeg2000). These functions have been used successfully in other areas, however. Elliptic Partial Differential Equations which model several processes in, for example, science and engineering, is one such field. This book, based on the author's course, 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.
Book Synopsis Numerical Analysis of Wavelet Methods by : A. Cohen
Download or read book Numerical Analysis of Wavelet Methods written by A. Cohen and published by Elsevier. This book was released on 2003-04-29 with total page 357 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 analysis of 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.
Book Synopsis Adaptive Wavelet Methods for Elliptic Stochastic Partial Differential Equations by : Petru A. Cioica
Download or read book Adaptive Wavelet Methods for Elliptic Stochastic Partial Differential Equations written by Petru A. Cioica and published by . This book was released on 2011 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Adaptive Wavelet Schwarz Methods for Nonlinear Elliptic Partial Differential Equations by : Dominik Lellek
Download or read book Adaptive Wavelet Schwarz Methods for Nonlinear Elliptic Partial Differential Equations written by Dominik Lellek and published by . This book was released on 2015 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Adaptive wavelet methods have recently proven to be a very powerful instrument for the numerical treatment of nonlinear partial differential equations. In many cases, these methods can be shown to converge with an optimal rate with respect to the degrees of freedom and in linear complexity. In this thesis, we couple such algorithms with nonlinear Schwarz domain decomposition techniques. With this approach, we can develop efficient parallel adaptive wavelet Schwarz methods for a class of nonlinear problems and prove their convergence and optimality. We support the theoretical findings with instructive numerical experiments. In addition, we present how these techniques can be applied to the stationary, incompressible Navier-Stokes equation. Furthermore, we couple the adaptive wavelet Schwarz methods with a Newton-type method.
Book Synopsis Wavelet Methods — Elliptic Boundary Value Problems and Control Problems by : Angela Kunoth
Download or read book Wavelet Methods — Elliptic Boundary Value Problems and Control Problems written by Angela Kunoth and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 150 pages. Available in PDF, EPUB and Kindle. Book excerpt: Diese Monographie spannt einen Bogen rund um die aktuelle Thematik Wavelets, um neueste Entwicklungen anhand aufeinander aufbauender Probleme darzustellen und das konzeptuelle Potenzial von Waveletmethoden für Partielle Differentialgleichungen zu demonstrieren.
Book Synopsis Wavelets, Multilevel Methods, and Elliptic PDEs by : M. Ainsworth
Download or read book Wavelets, Multilevel Methods, and Elliptic PDEs written by M. Ainsworth and published by Oxford University Press. This book was released on 1997 with total page 320 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book contains the Proceedings of the seventh EPSRC Numerical Analysis Summer School, held in 1996. Five major topics in numerical analysis are treated by world experts at a level which should be suitable for first year graduate students and experienced researchers alike, assuming onlythe knowledge acquired from a first degree in mathematics or in a scientific discipline with significant mathematical content. Often researchers need to obtain an up-to-date picture of work in an area with a substantial literature, either to avoid reproducing work which is already done, or to applyto their own research in a different subject. This book avoids the need to trawl through the literature by presenting important recent results together with references to all the main papers. Each contributor reviews the state of the art in his area, presenting new and often hitherto unpublishedmaterial.
Book Synopsis Partial Differential Equations by : D. Sloan
Download or read book Partial Differential Equations written by D. Sloan and published by Elsevier. This book was released on 2012-12-02 with total page 480 pages. Available in PDF, EPUB and Kindle. Book excerpt: /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.
Book Synopsis Wavelets Theory and Its Applications by : Mani Mehra
Download or read book Wavelets Theory and Its Applications written by Mani Mehra and published by Springer. This book was released on 2018-11-03 with total page 185 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides comprehensive information on the conceptual basis of wavelet theory and it applications. Maintaining an essential balance between mathematical rigour and the practical applications of wavelet theory, the book is closely linked to the wavelet MATLAB toolbox, which is accompanied, wherever applicable, by relevant MATLAB codes. The book is divided into four parts, the first of which is devoted to the mathematical foundations. The second part offers a basic introduction to wavelets. The third part discusses wavelet-based numerical methods for differential equations, while the last part highlights applications of wavelets in other fields. The book is ideally suited as a text for undergraduate and graduate students of mathematics and engineering.
Book Synopsis Wavelet Methods for Solving Partial Differential Equations and Fractional Differential Equations by : Santanu Saha Ray
Download or read book Wavelet Methods for Solving Partial Differential Equations and Fractional Differential Equations written by Santanu Saha Ray and published by CRC Press. This book was released on 2018-01-12 with total page 273 pages. Available in PDF, EPUB and Kindle. Book excerpt: The main focus of the book is to implement wavelet based transform methods for solving problems of fractional order partial differential equations arising in modelling real physical phenomena. It explores analytical and numerical approximate solution obtained by wavelet methods for both classical and fractional order partial differential equations.
Book Synopsis Numerical Solution of Elliptic Differential Equations by Reduction to the Interface by : Boris N. Khoromskij
Download or read book Numerical Solution of Elliptic Differential Equations by Reduction to the Interface written by Boris N. Khoromskij and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 304 pages. Available in PDF, EPUB and Kindle. Book excerpt: During the last decade essential progress has been achieved in the analysis and implementation of multilevel/rnultigrid and domain decomposition methods to explore a variety of real world applications. An important trend in mod ern numerical simulations is the quick improvement of computer technology that leads to the well known paradigm (see, e. g. , [78,179]): high-performance computers make it indispensable to use numerical methods of almost linear complexity in the problem size N, to maintain an adequate scaling between the computing time and improved computer facilities as N increases. In the h-version of the finite element method (FEM), the multigrid iteration real izes an O(N) solver for elliptic differential equations in a domain n c IRd d with N = O(h- ) , where h is the mesh parameter. In the boundary ele ment method (BEM) , the traditional panel clustering, fast multi-pole and wavelet based methods as well as the modern hierarchical matrix techniques are known to provide the data-sparse approximations to the arising fully populated stiffness matrices with almost linear cost O(Nr log?Nr), where 1 d Nr = O(h - ) is the number of degrees of freedom associated with the boundary. The aim of this book is to introduce a wider audience to the use of a new class of efficient numerical methods of almost linear complexity for solving elliptic partial differential equations (PDEs) based on their reduction to the interface.
Book Synopsis Sparse Grids and Applications - Munich 2012 by : Jochen Garcke
Download or read book Sparse Grids and Applications - Munich 2012 written by Jochen Garcke and published by Springer Science & Business Media. This book was released on 2014-04-11 with total page 345 pages. Available in PDF, EPUB and Kindle. Book excerpt: Sparse grids have gained increasing interest in recent years for the numerical treatment of high-dimensional problems. Whereas classical numerical discretization schemes fail in more than three or four dimensions, sparse grids make it possible to overcome the “curse” of dimensionality to some degree, extending the number of dimensions that can be dealt with. This volume of LNCSE collects the papers from the proceedings of the second workshop on sparse grids and applications, demonstrating once again the importance of this numerical discretization scheme. The selected articles present recent advances on the numerical analysis of sparse grids as well as efficient data structures, and the range of applications extends to uncertainty quantification settings and clustering, to name but a few examples.
Book Synopsis Mathematical Modelling, Optimization, Analytic and Numerical Solutions by : Pammy Manchanda
Download or read book Mathematical Modelling, Optimization, Analytic and Numerical Solutions written by Pammy Manchanda and published by Springer Nature. This book was released on 2020-02-04 with total page 431 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book discusses a variety of topics related to industrial and applied mathematics, focusing on wavelet theory, sampling theorems, inverse problems and their applications, partial differential equations as a model of real-world problems, computational linguistics, mathematical models and methods for meteorology, earth systems, environmental and medical science, and the oil industry. It features papers presented at the International Conference in Conjunction with 14th Biennial Conference of ISIAM, held at Guru Nanak Dev University, Amritsar, India, on 2–4 February 2018. The conference has emerged as an influential forum, bringing together prominent academic scientists, experts from industry, and researchers. The topics discussed include Schrodinger operators, quantum kinetic equations and their application, extensions of fractional integral transforms, electrical impedance tomography, diffuse optical tomography, Galerkin method by using wavelets, a Cauchy problem associated with Korteweg–de Vries equation, and entropy solution for scalar conservation laws. This book motivates and inspires young researchers in the fields of industrial and applied mathematics.
Book Synopsis Variational Techniques for Elliptic Partial Differential Equations by : Francisco J. Sayas
Download or read book Variational Techniques for Elliptic Partial Differential Equations written by Francisco J. Sayas and published by CRC Press. This book was released on 2019-01-16 with total page 492 pages. Available in PDF, EPUB and Kindle. Book excerpt: Variational Techniques for Elliptic Partial Differential Equations, intended for graduate students studying applied math, analysis, and/or numerical analysis, provides the necessary tools to understand the structure and solvability of elliptic partial differential equations. Beginning with the necessary definitions and theorems from distribution theory, the book gradually builds the functional analytic framework for studying elliptic PDE using variational formulations. Rather than introducing all of the prerequisites in the first chapters, it is the introduction of new problems which motivates the development of the associated analytical tools. In this way the student who is encountering this material for the first time will be aware of exactly what theory is needed, and for which problems. Features A detailed and rigorous development of the theory of Sobolev spaces on Lipschitz domains, including the trace operator and the normal component of vector fields An integration of functional analysis concepts involving Hilbert spaces and the problems which can be solved with these concepts, rather than separating the two Introduction to the analytical tools needed for physical problems of interest like time-harmonic waves, Stokes and Darcy flow, surface differential equations, Maxwell cavity problems, etc. A variety of problems which serve to reinforce and expand upon the material in each chapter, including applications in fluid and solid mechanics
