Sobolev Gradients and Differential Equations: 1670 - Brossura

Neuberger, John

Brossura

ISBN 10: 3642040403 ISBN 13: 9783642040405

Casa editrice: Springer, 2009

Riassunto

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.

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Contenuti

Several Gradients.- Comparison of Two Gradients.- Continuous Steepest Descent in Hilbert Space: Linear Case.- Continuous Steepest Descent in Hilbert Space: Nonlinear Case.- Orthogonal Projections, Adjoints and Laplacians.- Introducing Boundary Conditions.- Newton's Method in the Context of Sobolev Gradients.- Finite Difference Setting: the Inner Product Case.- Sobolev Gradients for Weak Solutions: Function Space Case.- Sobolev Gradient in Non-inner Product Spaces: Introduction.- The Superconductivity Equations of Ginzburg-Landau.- Minimal Surfaces.- Flow Problems and Non-inner Product Sobolev Spaces.- Foliations as a Guide to Boundary Conditions.- Some Related Iterative Methods for Differential Equations.- A Related Analytic Iteration Method.- Steepest Descent for Conservation Equations.- A Sample Computer Code with Notes.- Bibliography.- Index.

Product Description

Book by Neuberger John

Dettagli dell'edizione

• Editore: Springer
• Data di pubblicazione: 2009
• Lingua: Inglese
• ISBN 10: 3642040403
• ISBN 13: 9783642040405
• Rilegatura: Copertina flessibile
• Numero edizione: 2
• Numero di pagine: 312
• Contatto del produttore: non disponibile
• Persona responsabile: CMN Printpool Inh. Mathis Neumann; https://www.also.com/ec/cms5/en_6000/6000/index.jsp; ; Friedrich-Karl-Str. 9; Hamburg; Germania