Ebook: Sobolev Gradients and Differential Equations

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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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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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.

Content:
Front Matter....Pages i-xiii
Several Gradients....Pages 1-4
Comparison of Two Gradients....Pages 5-13
Continuous Steepest Descent in Hilbert Space: Linear Case....Pages 15-17
Continuous Steepest Descent in Hilbert Space: Nonlinear Case....Pages 19-34
Orthogonal Projections, Adjoints and Laplacians....Pages 35-51
Ordinary Differential Equations and Sobolev Gradients....Pages 53-55
Convexity and Gradient Inequalities....Pages 57-61
Boundary and Supplementary Conditions....Pages 63-78
Continuous Newton’s Method....Pages 79-83
More About Finite Differences....Pages 85-97
Sobolev Gradients for Variational Problems....Pages 99-102
An Introduction to Sobolev Gradients in Non-Inner Product Spaces....Pages 103-107
Singularities and a Simple Ginzburg-Landau Functional....Pages 109-111
The Superconductivity Equations of Ginzburg-Landau....Pages 113-121
Tricomi Equation: A Case Study....Pages 123-127
Minimal Surfaces....Pages 129-145
Flow Problems and Non-Inner Product Sobolev Spaces....Pages 147-152
An Alternate Approach to Time-dependent PDEs....Pages 153-158
Foliations and Supplementary Conditions I....Pages 159-169
Foliations and Supplementary Conditions II....Pages 171-175
Some Related Iterative Methods for Differential Equations....Pages 177-186
An Analytic Iteration Method....Pages 187-191
Steepest Descent for Conservation Equations....Pages 193-194
Code for an Ordinary Differential Equation....Pages 195-197
Geometric Curve Modeling with Sobolev Gradients....Pages 199-208
Numerical Differentiation, Sobolev Gradients....Pages 209-223
Steepest Descent and Newton’s Method and Elliptic PDE....Pages 225-238
Ginzburg-Landau Separation Problems....Pages 239-243
Numerical Preconditioning Methods for Elliptic PDEs....Pages 245-257
More Results on Sobolev Gradient Problems....Pages 259-271
Notes and Suggestions for Future Work....Pages 273-276
Back Matter....Pages 277-289

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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.

Content:
Front Matter....Pages i-xiii
Several Gradients....Pages 1-4
Comparison of Two Gradients....Pages 5-13
Continuous Steepest Descent in Hilbert Space: Linear Case....Pages 15-17
Continuous Steepest Descent in Hilbert Space: Nonlinear Case....Pages 19-34
Orthogonal Projections, Adjoints and Laplacians....Pages 35-51
Ordinary Differential Equations and Sobolev Gradients....Pages 53-55
Convexity and Gradient Inequalities....Pages 57-61
Boundary and Supplementary Conditions....Pages 63-78
Continuous Newton’s Method....Pages 79-83
More About Finite Differences....Pages 85-97
Sobolev Gradients for Variational Problems....Pages 99-102
An Introduction to Sobolev Gradients in Non-Inner Product Spaces....Pages 103-107
Singularities and a Simple Ginzburg-Landau Functional....Pages 109-111
The Superconductivity Equations of Ginzburg-Landau....Pages 113-121
Tricomi Equation: A Case Study....Pages 123-127
Minimal Surfaces....Pages 129-145
Flow Problems and Non-Inner Product Sobolev Spaces....Pages 147-152
An Alternate Approach to Time-dependent PDEs....Pages 153-158
Foliations and Supplementary Conditions I....Pages 159-169
Foliations and Supplementary Conditions II....Pages 171-175
Some Related Iterative Methods for Differential Equations....Pages 177-186
An Analytic Iteration Method....Pages 187-191
Steepest Descent for Conservation Equations....Pages 193-194
Code for an Ordinary Differential Equation....Pages 195-197
Geometric Curve Modeling with Sobolev Gradients....Pages 199-208
Numerical Differentiation, Sobolev Gradients....Pages 209-223
Steepest Descent and Newton’s Method and Elliptic PDE....Pages 225-238
Ginzburg-Landau Separation Problems....Pages 239-243
Numerical Preconditioning Methods for Elliptic PDEs....Pages 245-257
More Results on Sobolev Gradient Problems....Pages 259-271
Notes and Suggestions for Future Work....Pages 273-276
Back Matter....Pages 277-289
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