Numerical Analysis and Optimization An Introduction to Mathematical Modelling and Numerical Simulation

by Allaire, Grégoire; Craig, Alan

• ISBN13: 9780199205226
• ISBN10: 0199205221
• eBook ISBN(s): 9780191525520
• Format: Paperback
• Copyright: 2007-07-19
• Publisher: Oxford University Press
• More Book Details

Summary

Based on the Author's teaching notes at the Ecole Polytechnique,iNumerical Analysis and Optimization/i familiarises students with existingmathematical models (often partial differential equations) and their methods ofnumerical solution and optimization. The role of modelling and scientificcomputing has increased dramatically over recent years, and new applications ofmathematical models have emerged in Biology, Environmental Science, Finance,Medicine, and Social Science, as well as the classical applications inChemistry, Mechanics and Physics.Including numerous exercises and examples, this is an ideal text for advancedundergraduates and graduates and researchers in Applied Mathematics,Engineering, Computer Science, and the Physical Sciences.

Table of Contents

Introduction	p. ix
Introduction to mathematical modelling and numerical singulation	p. 1
General introduction	p. 1
An example of modelling	p. 2
Some classical models	p. 9
The heat flow equation	p. 9
The wave equation	p. 9
The Laplacian	p. 11
Schrodinger's equation	p. 12
The Lame system	p. 12
The Stokes system	p. 13
The plate equations	p. 13
Numerical calculation by finite differences	p. 14
Principles of the method	p. 14
Numerical results for the heat flow equation	p. 17
Numerical results for the advection equation	p. 21
Remarks on mathematical models	p. 25
The idea of a well-posed problem	p. 25
Classification of PDEs	p. 28
Finite difference method	p. 31
Introduction	p. 31
Finite differences for the heat equation	p. 32
Various examples of schemes	p. 32
Consistency and accuracy	p. 35
Stability and Fourier analysis	p. 36
Convergence of the schemes	p. 42
Multilevel schemes	p. 44
The multidimensional case	p. 46
Other models	p. 51
Advection equation	p. 51
Wave equation	p. 59
Variational formulation of elliptic problems	p. 65
Generalities	p. 65
Introduction	p. 65
Classical formulation	p. 66
The case of a space of one dimension	p. 67
Variational approach	p. 68
Green's formulas	p. 68
Variational formulation	p. 71
Lax-Milgram theory	p. 73
Abstract framework	p. 73
Application to the Laplacian	p. 76
Sobolev spaces	p. 79
Introduction and warning	p. 79
Square integrable functions and weak differentiation	p. 80
Some results from integration	p. 80
Weak differentiation	p. 81
Definition and principal properties	p. 84
The space H[superscript 1][ohm]	p. 84
The space [Characters not reproducible] [ohm]	p. 88
Traces and Green's formulas	p. 89
A compactness result	p. 94
The spaces H[superscript m] [ohm]	p. 96
Some useful extra results	p. 98
Proof of the density theorem 4.3.5	p. 98
The space H(div)	p. 101
The spaces W[superscript m,p] [ohm]	p. 102
Duality	p. 103
Link with distributions	p. 105
Mathematical study of elliptic problems	p. 109
Introduction	p. 109
Study of the Laplacian	p. 109
Dirichlet boundary conditions	p. 109
Neumann boundary conditions	p. 116
Variable coefficients	p. 123
Qualitative properties	p. 126
Solution of other models	p. 136
System of linear elasticity	p. 136
Stokes equations	p. 144
Finite element method	p. 149
Variational approximation	p. 149
Introduction	p. 149
Genera] internal approximation	p. 150
Galerkin method	p. 153
Finite element method (general principles)	p. 153
Finite elements in N = 1 dimension	p. 154
P[subscript 1] finite elements	p. 154
Convergence and error estimation	p. 159
P[subscript 2] finite elements	p. 163
Qualitative properties	p. 165
Hermite finite elements	p. 168
Finite elements in N [greater than or equal] 2 dimensions	p. 171
Triangular finite elements	p. 171
Convergence and error estimation	p. 184
Rectangular finite elements	p. 191
Finite elements for the Stokes problem	p. 195
Visualization of the numerical results	p. 201
Eigenvalue problems	p. 205
Motivation and examples	p. 205
Introduction	p. 205
Solution of nonstationary problems	p. 206
Spectral theory	p. 208
Generalities	p. 209
Spectral decomposition of a compact operator	p. 210
Eigenvalues of an elliptic problem	p. 213
Variational problem	p. 213
Eigenvalues of the Laplacian	p. 218
Other models	p. 221
Numerical methods	p. 224
Discretization by finite elements	p. 224
Convergence and error estimates	p. 227
Evolution problems	p. 231
Motivation and examples	p. 231
Introduction	p. 231
Modelling and examples of parabolic equations	p. 232
Modelling and examples of hyperbolic equations	p. 233
Existence and uniqueness in the parabolic case	p. 234
Variational formulation	p. 234
A general result	p. 236
Applications	p. 241
Existence and uniqueness in the hyperbolic case	p. 246
Variational formulation	p. 246
A general result	p. 247
Applications	p. 249
Qualitative properties in the parabolic case	p. 253
Asymptotic behaviour	p. 253
The maximum principle	p. 255
Propagation at infinite velocity	p. 256
Regularity and regularizing effect	p. 257
Heat equation in the entire space	p. 259
Qualitative properties in the hyperbolic case	p. 261
Reversibility in time	p. 261
Asymptotic behaviour and equipartition of energy	p. 262
Finite velocity of propagation	p. 263
Numerical methods in the parabolic case	p. 264
Semidiscretization in space	p. 264
Total discretization in space-time	p. 266
Numerical methods in the hyperbolic case	p. 269
Semidiscretization in space	p. 270
Total discretization in space-time	p. 271
Introduction to optimization	p. 277
Motivation and examples	p. 277
Introduction	p. 277
Examples	p. 278
Definitions and notation	p. 284
Optimization in finite dimensions	p. 285
Existence at a minimum in infinite dimensions	p. 287
Examples of nonexistence	p. 287
Convex analysis	p. 289
Existence results	p. 292
Optimality conditions and algorithms	p. 297
Generalities	p. 297
Introduction	p. 297
Differentiability	p. 298
Optimality conditions	p. 303
Euler inequalities and convex constraints	p. 303
Lagrange multipliers	p. 306
Saddle point, Kuhn-Tucker theorem, duality	p. 317
Saddle point	p. 317
The Kuhn-Tucker theorem	p. 318
Duality	p. 320
Applications	p. 323
Dual or complementary energy	p. 323
Optimal command	p. 326
Optimization of distributed systems	p. 330
Numerical algorithms	p. 332
Introduction	p. 332
Gradient algorithms (case without constraints)	p. 333
Gradient algorithms (case with constraints)	p. 336
Newton's method	p. 342
Methods of operational research	p. 347
Introduction	p. 347
Linear programming	p. 348
Definitions and properties	p. 348
The simplex algorithm	p. 353
Interior point algorithms	p. 357
Duality	p. 358
Integer polyhedra	p. 361
Extreme points of compact convex sets	p. 362
Totally unimodular matrices	p. 364
Flow problems	p. 368
Dynamic programming	p. 371
Bellman's optimality principle	p. 372
Finite horizon problem	p. 372
Minimum cost path, or optimal stopping, problem	p. 375
Greedy algorithms	p. 380
General points about greedy methods	p. 380
Kruskal's algorithm for the minimum spanning tree problem	p. 381
Separation and relaxation	p. 383
Separation and evaluation (branch and bound)	p. 383
Relaxation of combinatorial problems	p. 388
Appendix Review of hilbert spaces	p. 399
Appendix Matrix Numerical Analysis	p. 405
Solution of linear systems	p. 405
Review of matrix norms	p. 406
Conditioning and stability	p. 409
Direct methods	p. 411
Iterative methods	p. 424
The conjugate gradient method	p. 428
Calculation of eigenvalues and eigenvectors	p. 435
The power method	p. 436
The Givens-Householder method	p. 438
The Lanczos method	p. 442
Index	p. 451
Index notations	p. 455
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