Lecture Notes in Mathematics

By: john neuberger

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Paperback | 1 December 2009 | Edition Number 2

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312 Pages

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Revised

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22.23 x 15.24 x 1.91

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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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Non-Fiction Mathematics Calculus & Mathematical Analysis Functional Analysis & Transforms Differential Calculus & Equations Numerical Analysis

Several Gradients	p. 1
Comparison of Two Gradients	p. 5
A Graphical Comparison Between two Gradients	p. 11
Continuous Steepest Descent in Hilbert Space: Linear Case	p. 15
Continuous Steepest Descent in Hilbert Space: Nonlinear Case	p. 19
Global Existence	p. 19
Gradient Inequality	p. 21
Work of Chill and Huang on Gradient Inequalities	p. 30
Higher Order Sobolev Spaces for Lower Order Problems	p. 31
Orthogonal Projections, Adjoints and Laplacians	p. 35
A Construction of a Sobolev Space	p. 35
A Formula of von Neumann	p. 38
Relationship Between Adjoints	p. 39
General Laplacians	p. 40
Extension of Projections Beyond Hilbert Spaces	p. 45
A Generalized Lax-Milgram Theorem	p. 46
Laplacians and Closed Linear Transformations	p. 48
Projections for Higher Order Sobolev Spaces	p. 51
Ordinary Differential Equations and Sobolev Gradients	p. 53
Convexity and Gradient Inequalities	p. 57
Boundary and Supplementary Conditions	p. 63
Introduction	p. 63
Orthogonal Projection onto a Null Space	p. 65
Projected Sobolev Gradients, Linear Case	p. 65
Projected Gradients, Nonlinear Case	p. 66
Explicit Form for a Projected Gradient	p. 66
Continuous vs Discrete Steepest Descent	p. 69
A Finite Dimensional Example for Adjoints	p. 70
Approximation of Projected Gradients	p. 73
An Example with Mixed Boundary Conditions	p. 76
Continuous Newton's Method	p. 79
Riemannian Metrics and a Nash-Moser Inverse Function Result	p. 79
Newton's Method from Optimization	p. 82
More About Finite Differences	p. 85
Finite Differences and Sobolev Gradients	p. 85
Supplementary Conditions Again	p. 88
Graphs and Sobolev Gradients	p. 90
Digression on Adjoints of Difference Operators	p. 92
A First Order Partial Differential Equation	p. 92
A Second Order Partial Differential Equation	p. 95
Sobolev Gradients for Variational Problems	p. 99
Minimizing Sequences	p. 99
Euler-Lagrange Equations	p. 100
Sobolev Gradient Approach	p. 100
An Introduction to Sobolev Gradients in Non-Inner Product Spaces	p. 103
Singularities and a Simple Ginzburg-Landau Functional	p. 109
The Superconductivity Equations of Ginzburg-Landau	p. 113
Introduction	p. 113
A GL Functional and Its Sobolev Gradient	p. 113
Finite Dimensional Emulation	p. 116
Numerical Results	p. 117
Tricomi Equation: A Case Study	p. 123
Numerical Simulation for Tricomi's Equation	p. 125
Experimenting With Boundary Conditions	p. 125
Minimal Surfaces	p. 129
Introduction	p. 129
Minimum Curve Length	p. 129
Minimal Surfaces	p. 132
Uniformly Parameterized Surfaces	p. 136
Numerical Methods and Test Results	p. 140
Conclusion	p. 145
Flow Problems and Non-Inner Product Sobolev Spaces	p. 147
Full Potential Equation	p. 147
Other Codes for Transonic Flow	p. 150
Transonic Flow Plots	p. 151
An Alternate Approach to Time-Dependent PDEs	p. 153
Introduction	p. 153
Least Squares Method	p. 155
Numerical Results	p. 156
Foliations and Supplementary Conditions I	p. 159
A Foliation Theorem	p. 159
A Linear Example	p. 168
Foliations and Supplementary Conditions II	p. 171
Semigroups on a Metric Space	p. 171
Application: Supplementary Condition Problem	p. 172
Computational Fantasy	p. 174
Some Related Iterative Methods for Differential Equations	p. 177
An Analytic Iteration Method	p. 187
Steepest Descent for Conservation Equations	p. 193
Code for an Ordinary Differential Equation	p. 195
Geometric Curve Modeling with Sobolev Gradients	p. 199
Introduction	p. 199
Minimum Curve-Length	p. 200
Discrete Minimum-Length Curves	p. 204
Test Results	p. 207
Numerical Differentiation, Sobolev Gradients	p. 209
Introduction	p. 209
The Functional H(q)	p. 212
Stability	p. 216
Steepest Descent Minimization	p. 217
Some Numerical Examples	p. 219
Related Inverse Problems	p. 221
Steepest Descent and Newton's Method and Elliptic PDE	p. 225
Introduction	p. 225
The Variational Formulation for PDE and PdE	p. 226
Algorithms	p. 230
The MPA and MMPA	p. 231
The GNGA	p. 232
Tangent-Augmented Newton's Method (tGNGA)	p. 233
The Secant Method	p. 233
Cylinder-Augmented Newton's Method (cGNGA)	p. 234
Some PDE and PdE Results	p. 235
Ginzburg-Landau Separation Problems	p. 239
Introduction	p. 239
Model A	p. 240
Weighted Gradients	p. 241
Model A'	p. 241
A Phase Separation Problem	p. 242
An Elasticity Problem	p. 243
Numerical Preconditioning Methods for Elliptic PDEs	p. 245
Introduction	p. 245
The Model Problem	p. 246
Condition Numbers of Nonlinear Operators	p. 247
Fixed Preconditioners: Sobolev Gradients with Fixed Inner Product	p. 248
Sobolev Gradients and Laplacian Preconditioners	p. 249
General Preconditioners as Weighted Sobolev Gradients	p. 251
Variable Preconditioners: Sobolev Gradients with Variable Inner Product	p. 252
Quasi-Newton Methods as Variable Steepest Descent	p. 252
Newton's Method as an Optimal Variable Steepest Descent	p. 254
Mixed Problems and Other Extensions	p. 256
More Results on Sobolev Gradient Problems	p. 259
Singular Boundary Value Problems	p. 259
Quantum Mechanical Calculations	p. 260
Dual Steepest Descent	p. 261
Optimal Embedding Constants for Sobolev Spaces	p. 262
Performance of Preconditioners and H-1 Methods	p. 262
Poisson-Boltzmann Equation	p. 263
Time Independent Navier-Stokes	p. 263
Steepest Descent and Hyperbolic Monge-Ampere Equations	p. 264
Application to Differential Algebraic Equations	p. 265
Control Theory and PDE	p. 266
An Elasticity Problem	p. 267
A Liquid Crystal Problem	p. 267
Applications to Functional Differential Equations	p. 268
More About Active Contours	p. 269
Another Solution Giving Nonlinear Projection	p. 270
Dynamics of Steepest Descent	p. 271
Aubry-Mather Theory and a Comparison Principle for a Sobolev Gradient Descent	p. 271
Notes and Suggestions for Future Work	p. 273
References	p. 277
Index	p. 287
Table of Contents provided by Ingram. All Rights Reserved.

Lecture Notes in Mathematics

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Paperback

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