Download or read book Sobolev Gradients and Differential Equations written by John Neuberger and published by Springer Science & Business Media. This book was released on 2009-12-01 with total page 287 pages. Available in PDF, EPUB and Kindle. Book excerpt: A Sobolev gradient of a real-valued functional on a Hilbert space is a gradient of that functional taken relative to an underlying Sobolev norm. This book shows how descent methods using such gradients allow a unified treatment of a wide variety of problems in differential equations. For discrete versions of partial differential equations, corresponding Sobolev gradients are seen to be vastly more efficient than ordinary gradients. In fact, descent methods with these gradients generally scale linearly with the number of grid points, in sharp contrast with the use of ordinary gradients. Aside from the first edition of this work, this is the only known account of Sobolev gradients in book form. Most of the applications in this book have emerged since the first edition was published some twelve years ago. What remains of the first edition has been extensively revised. There are a number of plots of results from calculations and a sample MatLab code is included for a simple problem. Those working through a fair portion of the material have in the past been able to use the theory on their own applications and also gain an appreciation of the possibility of a rather comprehensive point of view on the subject of partial differential equations.

Download or read book Sobolev Gradient Methods written by Nauman Raza and published by LAP Lambert Academic Publishing. This book was released on 2010-08-01 with total page 116 pages. Available in PDF, EPUB and Kindle. Book excerpt: Sobolev gradient methods resolve numerical difficulties in approximating solutions to differential equations and minima of error and energy functionals by construction of inner product spaces that one suitable for the problem at hand. The great efficiency achieved by setting the problem in right Sobolev space, makes steepest descent methods applicable to wide variety of problems. In this monograph, applications of Sobolev gradient methods in finite-difference and finite-element settings are considered for minimization of energy functionals, soliton solutions of the nonlinear Schrodinger equation, and pulse propagation through a fiber optic cable. For each problem, the practical application of the principle of selecting an appropriate Sobolev space setting is demonstrated. The advantages of the Sobolev gradient approach in efficiency and simplicity of implementation are shown. Engineers and computational physicists will find a clear description of the numerical method allowing immediate applications to problems of their interest.

Download or read book Sobolev Gradients and Differential Equations written by john neuberger and published by Springer. This book was released on 2009-11-10 with total page 287 pages. Available in PDF, EPUB and Kindle. Book excerpt: A Sobolev gradient of a real-valued functional on a Hilbert space is a gradient of that functional taken relative to an underlying Sobolev norm. This book shows how descent methods using such gradients allow a unified treatment of a wide variety of problems in differential equations. For discrete versions of partial differential equations, corresponding Sobolev gradients are seen to be vastly more efficient than ordinary gradients. In fact, descent methods with these gradients generally scale linearly with the number of grid points, in sharp contrast with the use of ordinary gradients. Aside from the first edition of this work, this is the only known account of Sobolev gradients in book form. Most of the applications in this book have emerged since the first edition was published some twelve years ago. What remains of the first edition has been extensively revised. There are a number of plots of results from calculations and a sample MatLab code is included for a simple problem. Those working through a fair portion of the material have in the past been able to use the theory on their own applications and also gain an appreciation of the possibility of a rather comprehensive point of view on the subject of partial differential equations.

Download or read book Sobolev Gradients and Differential Equations written by john neuberger and published by Springer. This book was released on 2006-11-13 with total page 150 pages. Available in PDF, EPUB and Kindle. Book excerpt: A Sobolev gradient of a real-valued functional is a gradient of that functional taken relative to the underlying Sobolev norm. This book shows how descent methods using such gradients allow a unified treatment of a wide variety of problems in differential equations. Equal emphasis is placed on numerical and theoretical matters. Several concrete applications are made to illustrate the method. These applications include (1) Ginzburg-Landau functionals of superconductivity, (2) problems of transonic flow in which type depends locally on nonlinearities, and (3) minimal surface problems. Sobolev gradient constructions rely on a study of orthogonal projections onto graphs of closed densely defined linear transformations from one Hilbert space to another. These developments use work of Weyl, von Neumann and Beurling.

Download or read book Sobolev Gradients and Differential Equations written by John W. Neuberger and published by . This book was released on 1997 with total page 164 pages. Available in PDF, EPUB and Kindle. Book excerpt: A Sobolev gradient of a real-valued functional is a gradient of that functional taken relative to the underlying Sobolev norm. This book shows how descent methods using such gradients allow a unified treatment of a wide variety of problems in differential equations. Equal emphasis is placed on numerical and theoretical matters. Several concrete applications are made to illustrate the method. These applications include (1) Ginzburg-Landau functionals of superconductivity, (2) problems of transonic flow in which type depends locally on nonlinearities, and (3) minimal surface problems. Sobolev gradient constructions rely on a study of orthogonal projections onto graphs of closed densely defined linear transformations from one Hilbert space to another. These developments use work of Weyl, von Neumann and Beurling.

Download or read book Sobolev Gradient Semi flows Applications to Nonlinear Problems written by Ramesh Karki and published by . This book was released on 2015 with total page 111 pages. Available in PDF, EPUB and Kindle. Book excerpt: We are interested in solving nonlinear pseudo-differential equations (in particular, partial differential equations as well) involving fractional powers of uniformly elliptic self-adjoint operators of order two with suitable smoothness conditions on the coefficients subject to given (Dirichlet or periodic) boundary conditions. Under the stronger assumptions, we are interested in studying solutions in a special class whose elements satisfy non-selfintersecting property and have bounded distance from a given hyperplane, since such solutions are the analogue for Aubrey-Mather sets for ODEs and leaves of minimal foliations or minimal laminations for PDEs. To solve such a PsiDE, we will start by introducing an energy type functional whose Euler-Lagrange equation is the pseudo-differential equation itself. As we seek to minimize this functional, we will introduce the Sobolev gradient of the functional as an element of a suitable Sobolev space and then we consider the gradient descent equation subject to appropriate initial and boundary conditions. The equilibrium solutions of this Sobolev gradient descent equation are the critical points we are looking for. Now the first step of our work will be to construct a semi-flow corresponding to the aforementioned initial-boundary value problem. So we will prove the existence, uniqueness, regularity, and comparison properties related to the semi-flow. Then the next step will be to analyze the convergence of this semi-flow to an equilibrium solution to this initial-boundary value problem. In our work, we will adapt two methods: analytical method and numerical method. We apply various analytical tools to establish the general results and numerical tools to study concrete solutions of particular pseudo-differential or partial differential equations.

Download or read book Using Gradient Descent Method to Solve Systems of Differential Equations Under to Sobolev Inner Product Space written by Jason A. Hatton and published by . This book was released on 2017 with total page 114 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Application of Sobolev Gradient to Poisson Boltzmann System written by Abdul Majid and published by LAP Lambert Academic Publishing. This book was released on 2012-07 with total page 144 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book offers an application of Sobolev gradient approach to Poisson Boltzmann system. A detailed description of Sobolev gradient method is given and its application is demonstrated on the Poisson Boltzmann system when there are large non-linearities and discontinuities in the coefficient functions. Poisson Boltzmann is a physical model that governs the electrostatic potential of macromolecules when immersed in solvent. It is shown that in some cases Sobolev gradient performs better in terms of efficiency than other existing fast methods such as multigrid and Newton's methods. The experiments' results are given in both finite element and finite difference settings. This book presents a fine blend of Functional Analysis, Numerical Analysis and Biophysics. It is the Ph.D. work that Dr. Abdul Majid completed under the supervision of Dr. Sultan Sial.

Download or read book Sobolev Gradient Flows and Image Processing written by Jeffrey William Calder and published by . This book was released on 2010 with total page 262 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this thesis we study Sobolev gradient flows for Perona-Malik style energy functionals and generalizations thereof. We begin with first order isotropic flows which are shown to be regularizations of the heat equation. We show that these flows are well-posed in the forward and reverse directions which yields an effective linear sharpening algorithm. We furthermore establish a number of maximum principles for the forward flow and show that edges are preserved for a finite period of time. We then go on to study isotropic Sobolev gradient flows with respect to higher order Sobolev metrics. As the Sobolev order is increased, we observe an increasing reluctance to destroy fine details and texture. We then consider Sobolev gradient flows for non-linear anisotropic diffusion functionals of arbitrary order. We establish existence, uniqueness and continuous dependence on initial data for a broad class of such equations. The well-posedness of these new anisotropic gradient flows opens the door to a wide variety of sharpening and diffusion techniques which were previously impossible under L2 gradient descent. We show how one can easily use this framework to design an anisotropic sharpening algorithm which can sharpen image features while suppressing noise. We compare our sharpening algorithm to the well-known shock filter and show that Sobolev sharpening produces natural looking images without the "staircasing" artifacts that plague the shock filter.

Download or read book Sobolev Spaces on Metric Measure Spaces written by Juha Heinonen and published by Cambridge University Press. This book was released on 2015-02-05 with total page 447 pages. Available in PDF, EPUB and Kindle. Book excerpt: This coherent treatment from first principles is an ideal introduction for graduate students and a useful reference for experts.

Download or read book Numerical Solution of Nonlinear Elliptic Problems Via Preconditioning Operators written by István Faragó and published by Nova Publishers. This book was released on 2002 with total page 424 pages. Available in PDF, EPUB and Kindle. Book excerpt: Numerical Solution of Nonlinear Elliptic Problems Via Preconditioning Operators - Theory & Applications

Download or read book Functional Analysis Sobolev Spaces and Partial Differential Equations written by Haim Brezis and published by Springer Science & Business Media. This book was released on 2010-11-02 with total page 600 pages. Available in PDF, EPUB and Kindle. Book excerpt: This textbook is a completely revised, updated, and expanded English edition of the important Analyse fonctionnelle (1983). In addition, it contains a wealth of problems and exercises (with solutions) to guide the reader. Uniquely, this book presents in a coherent, concise and unified way the main results from functional analysis together with the main results from the theory of partial differential equations (PDEs). Although there are many books on functional analysis and many on PDEs, this is the first to cover both of these closely connected topics. Since the French book was first published, it has been translated into Spanish, Italian, Japanese, Korean, Romanian, Greek and Chinese. The English edition makes a welcome addition to this list.

Download or read book Variational Analysis in Sobolev and BV Spaces written by Hedy Attouch and published by SIAM. This book was released on 2014-10-02 with total page 794 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume is an excellent guide for anyone interested in variational analysis, optimization, and PDEs. It offers a detailed presentation of the most important tools in variational analysis as well as applications to problems in geometry, mechanics, elasticity, and computer vision. This second edition covers several new topics: new section on capacity theory and elements of potential theory now includes the concepts of quasi-open sets and quasi-continuity; increased number of examples in the areas of linearized elasticity system, obstacles problems, convection-diffusion, and semilinear equations; new section on mass transportation problems and the Kantorovich relaxed formulation of the Monge problem; new subsection on stochastic homogenization establishes the mathematical tools coming from ergodic theory; and an entirely new and comprehensive chapter (17) devoted to gradient flows and the dynamical approach to equilibria. The book is intended for Ph.D. students, researchers, and practitioners who want to approach the field of variational analysis in a systematic way.

Download or read book Library of Congress Subject Headings written by Library of Congress. Cataloging Policy and Support Office and published by . This book was released on 2009 with total page 1924 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Library of Congress Subject Headings written by Library of Congress and published by . This book was released on 2004 with total page 1480 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Biomedical Image Registration written by Bernd Fischer and published by Springer. This book was released on 2010-07-05 with total page 292 pages. Available in PDF, EPUB and Kindle. Book excerpt: Welcome to the proceedings of the 4th Workshop on Biomedical Image R- istration (WBIR). Previous WBIRs took place in Bled, Slovenia (1999), at the UniversityofPennsylvania,USA(2003)andinUtrecht,TheNetherlands(2006). This year, WBIR was hosted by the Institute Mathematics and Image Proce- ing and the Fraunhofer Project Group on Image Registration and it was held in Lub ̈ eck, Germany. It provided the opportunity to bring together researchers from all over the world to discuss some of the most recent advances in image registration and its applications. We had an excellent collection of papers that were reviewed by at least three reviewers each from a 35-member Program Committee assembled from a wor- wide community of registration experts. This year 17 papers were accepted for oral presentation, while another 7 papers were accepted as poster papers. We believe all of the conference papers were of excellent quality. Registration is a fundamental task in image processing used to match two or more pictures taken, for example, at di?erent times, from di?erent sensors, or from di?erent viewpoints. Establishing the correspondence of structures within medical images is fundamental to diagnosis, treatment planning, and surgical guidance. The conference papers address state-of-the-art techniques for prov- ing reliable and e?cient registration techniques, thereby imposing relationships between speci?c application areas and appropriate registration schemes. We are grateful to all those who contributed to the success of WBIR 2010.

Download or read book Scale Space and Variational Methods in Computer Vision written by Fiorella Sgallari and published by Springer Science & Business Media. This book was released on 2007-07-23 with total page 934 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the refereed proceedings of the First International Conference on Scale Space Methods and Variational Methods in Computer Vision, SSVM 2007, emanated from the joint edition of the 4th International Workshop on Variational, Geometric and Level Set Methods in Computer Vision, VLSM 2007 and the 6th International Conference on Scale Space and PDE Methods in Computer Vision, Scale-Space 2007, held in Ischia Italy, May/June 2007.