000 04298nam a22004815i 4500
001 978-3-642-04041-2
003 DE-He213
005 20160624101836.0
007 cr nn 008mamaa
008 100301s2010 gw | s |||| 0|eng d
020 _a9783642040412
_9978-3-642-04041-2
024 7 _a10.1007/978-3-642-04041-2
_2doi
050 4 _aQA370-380
072 7 _aPBKJ
_2bicssc
072 7 _aMAT007000
_2bisacsh
082 0 4 _a515.353
_223
100 1 _aNeuberger, J.W.
_eauthor.
245 1 0 _aSobolev Gradients and Differential Equations
_h[electronic resource] /
_cby J.W. Neuberger.
260 1 _aBerlin, Heidelberg :
_bSpringer Berlin Heidelberg,
_c2010.
264 1 _aBerlin, Heidelberg :
_bSpringer Berlin Heidelberg,
_c2010.
300 _aXIII, 289p.
_bonline resource.
336 _atext
_btxt
_2rdacontent
337 _acomputer
_bc
_2rdamedia
338 _aonline resource
_bcr
_2rdacarrier
347 _atext file
_bPDF
_2rda
490 1 _aLecture Notes in Mathematics,
_x0075-8434 ;
_v1670
505 0 _aSeveral 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 -- Ordinary Differential Equations and Sobolev Gradients -- Convexity and Gradient Inequalities -- Boundary and Supplementary Conditions -- Continuous Newton’s Method -- More About Finite Differences -- Sobolev Gradients for Variational Problems -- An Introduction to Sobolev Gradients in Non-Inner Product Spaces -- Singularities and a Simple Ginzburg-Landau Functional -- The Superconductivity Equations of Ginzburg-Landau -- Tricomi Equation: A Case Study -- Minimal Surfaces -- Flow Problems and Non-Inner Product Sobolev Spaces -- An Alternate Approach to Time-dependent PDEs -- Foliations and Supplementary Conditions I -- Foliations and Supplementary Conditions II -- Some Related Iterative Methods for Differential Equations -- An Analytic Iteration Method -- Steepest Descent for Conservation Equations -- Code for an Ordinary Differential Equation -- Geometric Curve Modeling with Sobolev Gradients -- Numerical Differentiation, Sobolev Gradients -- Steepest Descent and Newton’s Method and Elliptic PDE -- Ginzburg-Landau Separation Problems -- Numerical Preconditioning Methods for Elliptic PDEs -- More Results on Sobolev Gradient Problems -- Notes and Suggestions for Future Work.
520 _aA 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.
650 0 _aMathematics.
650 0 _aDifferential equations, partial.
650 0 _aNumerical analysis.
650 1 4 _aMathematics.
650 2 4 _aPartial Differential Equations.
650 2 4 _aNumerical Analysis.
710 2 _aSpringerLink (Online service)
773 0 _tSpringer eBooks
776 0 8 _iPrinted edition:
_z9783642040405
786 _dSpringer
830 0 _aLecture Notes in Mathematics,
_x0075-8434 ;
_v1670
856 4 0 _uhttp://dx.doi.org/10.1007/978-3-642-04041-2
942 _2EBK1926
_cEBK
999 _c31220
_d31220