Sobolev gradients and differential equations - 2nd Ed.

By: Neuberger, J. WMaterial type: TextTextSeries: Lecture Notes in Mathematics ; 1670Publication details: New York: Springer, [c1997]Edition: 2nd edDescription: 289 pISBN: 9783642040405LOC classification: QA3
Contents:
1. Several Gradients 2. Comparison of Two Gradients 3. Continuous Steepest Descent in Hilbert Space: Linear Case 4. Continuous Steepest Descent in Hilbert Space: Nonlinear Case 5. Orthogonal Projections, Adjoints and Laplacians 6. Ordinary Differential Equations and Sobolev Gradients 7. Convexity and Gradient Inequalities 8. Boundary and Supplementary Conditions 9. Continuous Newton’s Method 10. More About Finite Differences 11. Sobolev Gradients for Variational Problems 12. An Introduction to Sobolev Gradients in Non-Inner Product Spaces 13. Singularities and a Simple Ginzburg-Landau Functional 14. The Superconductivity Equations of Ginzburg-Landau 15. Tricomi Equation: A Case Study 16. Minimal Surfaces 17. Flow Problems and Non-Inner Product Sobolev Spaces 18. An Alternate Approach to Time-dependent PDEs 19. Foliations and Supplementary Conditions I 20. Foliations and Supplementary Conditions II 21. Some Related Iterative Methods for Differential Equations 22. An Analytic Iteration Method 23. Steepest Descent for Conservation Equations 24. Code for an Ordinary Differential Equation 25. Geometric Curve Modeling with Sobolev Gradients 26. Numerical Differentiation, Sobolev Gradients 27. Steepest Descent and Newton’s Method and Elliptic PDE 28. Ginzburg-Landau Separation Problems 29. Numerical Preconditioning Methods for Elliptic PDEs 30. More Results on Sobolev Gradient Problems 31. Notes and Suggestions for Future Work
Summary: 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.
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1. Several Gradients
2. Comparison of Two Gradients
3. Continuous Steepest Descent in Hilbert Space: Linear Case
4. Continuous Steepest Descent in Hilbert Space: Nonlinear Case
5. Orthogonal Projections, Adjoints and Laplacians
6. Ordinary Differential Equations and Sobolev Gradients
7. Convexity and Gradient Inequalities
8. Boundary and Supplementary Conditions
9. Continuous Newton’s Method
10. More About Finite Differences
11. Sobolev Gradients for Variational Problems
12. An Introduction to Sobolev Gradients in Non-Inner Product Spaces
13. Singularities and a Simple Ginzburg-Landau Functional
14. The Superconductivity Equations of Ginzburg-Landau
15. Tricomi Equation: A Case Study
16. Minimal Surfaces
17. Flow Problems and Non-Inner Product Sobolev Spaces
18. An Alternate Approach to Time-dependent PDEs
19. Foliations and Supplementary Conditions I
20. Foliations and Supplementary Conditions II
21. Some Related Iterative Methods for Differential Equations
22. An Analytic Iteration Method
23. Steepest Descent for Conservation Equations
24. Code for an Ordinary Differential Equation
25. Geometric Curve Modeling with Sobolev Gradients
26. Numerical Differentiation, Sobolev Gradients
27. Steepest Descent and Newton’s Method and Elliptic PDE
28. Ginzburg-Landau Separation Problems
29. Numerical Preconditioning Methods for Elliptic PDEs
30. More Results on Sobolev Gradient Problems
31. Notes and Suggestions for Future Work

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.

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