rdf-syntax-grammar
mods:
  schema location: http://www.loc.gov/mods/v3 http://www.loc.gov/standards/mods/v3/mods-3-2.xsd
  title info:
    title: Sobolev Gradients and Differential Equations
  name:
    name part: Neuberger, J.W.
    role:
      role term: creator
    role:
      role term: author.
  name:
    name part: SpringerLink (Online service)
  type of resource: text
  origin info:
    place:
      place term: gw
    date issued: 2010
    issuance: monographic
  language:
    language term: eng
  physical description:
    form: electronic
    form: electronic resource
    reformatting quality: access
    form: remote
    extent: XIII, 289 p. online resource.
  abstract: 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.
  table of contents: 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 -- Ordinary Differential Equations and Sobolev Gradients -- Convexity and Gradient Inequalities -- Boundary and Supplementary Conditions -- Continuous Newton#x2019;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#x2019;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.
  note: by J.W. Neuberger.
  subject:
    topic: Mathematics
  subject:
    topic: Mathematical analysis
  subject:
    topic: Analysis (Mathematics)
  subject:
    topic: Partial differential equations
  subject:
    topic: Numerical analysis
  subject:
    topic: Mathematics
  subject:
    topic: Analysis
  subject:
    topic: Partial Differential Equations
  subject:
    topic: Numerical Analysis
  classification: QA299.6-433
  classification: 515
  related item:
    title info:
      title: Springer eBooks
  related item:
    title info:
      title: Lecture Notes in Mathematics, 1670
  identifier: 9783642040412
  identifier: http://dx.doi.org/10.1007/978-3-642-04041-2
  record info:
    record creation date: 100301
    record change date: 20151204175902.0
    record identifier: 978-3-642-04041-2