Numerical analysis and optimization: an introduction to mathematical modelling and numerical simulation
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English French |
| Veröffentlicht: |
Oxford
Oxford University Press
2007
|
| Ausgabe: | 1. publ. |
| Schriftenreihe: | Numerical mathematics and scientific computation
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Includes bibliographical references (p. 447-449) and index |
| Beschreibung: | xvi, 455 p. ill. 24 cm |
| ISBN: | 0199205213 0199205221 9780199205219 9780199205226 |
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| 240 | 1 | 0 | |a Analyse numérique et optimisation |
| 245 | 1 | 0 | |a Numerical analysis and optimization |b an introduction to mathematical modelling and numerical simulation |c Grégoire Allaire ; translated by Alan Craig |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a Oxford |b Oxford University Press |c 2007 | |
| 300 | |a xvi, 455 p. |b ill. |c 24 cm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Numerical mathematics and scientific computation | |
| 500 | |a Includes bibliographical references (p. 447-449) and index | ||
| 546 | |a Translated from the French. | ||
| 650 | 4 | |a Mathematisches Modell | |
| 650 | 4 | |a Numerical analysis | |
| 650 | 4 | |a Mathematical optimization | |
| 650 | 4 | |a Mathematical models | |
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Datensatz im Suchindex
| _version_ | 1804138627298492416 |
|---|---|
| adam_text | Contents
1
Introduction
ix
1
Introduction
to mathematical modelling and numerical simulation
1
1.1
General introduction
................................ 1
1.2
An example of modelling
.............................. 2
1.3
Some classical models
............................... 9
1.3.1
The heat flow equation
.......................... 9
1.3.2
The wave equation
............................. 9
1.3.3
The Laplacian
............................... 11
1.3.4 Schrödinger s
equation
........................... 12
1.3.5
The
Lamé
system
.............................. 12
1.3.6
The Stokes system
............................. 13
1.3.7
The plate equations
............................ 13
1.4
Numerical calculation by finite differences
.................... 14
1.4.1
Principles of the method
.......................... 14
1.4.2
Numerical results for the heat flow equation
............... 17
1.4.3
Numerical results for the advection equation
.............. 21
1.5
Remarks on mathematical models
......................... 25
1.5.1
The idea of a well-posed problem
..................... 25
1.5.2
Classification of PDEs
........................... 28
2
Finite difference method
31
2.1
Introduction
..................................... 31
2.2
Finite differences for the heat equation
...................... 32
2.2.1
Various examples of schemes
....................... 32
2.2.2
Consistency and accuracy
......................... 35
2.2.3
Stability and Fourier analysis
....................... 36
2.2.4
Convergence of the schemes
........................ 42
2.2.5
Multilevel schemes
............................. 44
2.2.6
The multidimensional case
......................... 46
2.3
Other models
.................................... 51
2.3.1
Advection equation
............................. 51
2.3.2
Wave equation
............................... 59
CONTENTS
Variational
formulation
of elliptic problems
65
3.1
Generalities
..................................... 65
3.1.1
Introduction
................................ 65
3.1.2
Classical formulation
............................ 66
3.1.3
The case of a space of one dimension
................... 67
3.2
Variational approach
................................ 68
3.2.1
Green s formulas
.............................. 68
3.2.2
Variational formulation
.......................... 71
3.3
Lax Milgram theory
................................ 73
3.3.1
Abstract framework
............................ 73
3.3.2
Application to the Laplacian
....................... 76
Sobolev spaces
79
4.1
Introduction and warning
............................. 79
4.2
Square
integrable
functions and weak differentiation
.............. 80
4.2.1
Some results from integration
....................... 80
4.2.2
Weak differentiation
............................ 81
4.3
Definition and principal properties
........................ 84
4.3.1
The space
tf^ü)
.............................. 84
4.3.2
The space
#¿(0).............................. 88
4.3.3
Traces and Green s formulas
....................... 89
4.3.4
A compactness result
............................ 94
4.3.5
The spaces
Нт{п)
............................. 96
4.4
Some useful extra results
.............................. 98
4.4.1
Proof of the density theorem
4.3.5.................... 98
4.4.2
The space H(div)
............................. 101
4.4.3
The spaces Wm p{U)
............................ 102
4.4.4
Duality
................................... 103
4.5
Link with distributions
............................... 105
Mathematical study of elliptic problems
109
5.1
Introduction
..................................... 109
5.2
Study of the Laplacian
............................... 109
5.2.1
Dirichlet boundary conditions
....................... 109
5.2.2
Neumann boundary conditions
...................... 116
5.2.3
Variable coefficients
............................ 123
5.2.4
Qualitative properties
........................... 126
5.3
Solution of other models
.............................. 136
5.3.1
System of linear elasticity
......................... 136
5.3.2
Stokes equations
.............................. 144
Finite element method
149
6.1
Variational approximation
............................. 149
6.1.1
Introduction
................................ 149
6.1.2
General internal approximation
...................... 150
6.1.3
Galerkin method
.............................. 153
6.1.4
Finite element method (general principles)
................ 153
6.2
Finite elements in
N = 1
dimension
....................... 154
CONTENTS
vii
6.2.1
Pi
ßnite
elements
............................. 154
6.2.2
Convergence and error estimation
..................... 159
6.2.3
P2 finite elements
.............................. 163
6.2.4
Qualitative properties
........................... 165
6.2.5
Hermite finite elements
.......................... 168
6.3
Finite elements in
N > 2
dimensions
....................... 171
6.3.1
Triangular finite elements
......................... 171
6.3.2
Convergence and error estimation
..................... 184
6.3.3
Rectangular finite elements
........................ 191
6.3.4
Finite elements for the Stokes problem
.................. 195
6.3.5
Visualization of the numerical results
................... 201
7
Eigenvalue problems
205
7.1
Motivation and examples
............................. 205
7.1.1
Introduction
................................ 205
7.1.2
Solution of nonstationary problems
.................... 206
7.2
Spectral theory
................................... 208
7.2.1
Generalities
................................. 209
7.2.2
Spectral decomposition of a compact operator
.............. 210
7.3
Eigenvalues of an elliptic problem
......................... 213
7.3.1
Variational problem
............................ 213
7.3.2
Eigenvalues of the Laplacian
....................... 218
7.3.3
Other models
................................ 221
7.4
Numerical methods
................................. 224
7.4.1
Discretization by finite elements
..................... 224
7.4.2
Convergence and error estimates
..................... 227
8
Evolution problems
231
8.1
Motivation and examples
............................. 231
8.1.1
Introduction
................................ 231
8.1.2
Modelling and examples of parabolic equations
............. 232
8.1.3
Modelling and examples of hyperbolic equations
............ 233
8.2
Existence and uniqueness in the parabolic case
................. 234
8.2.1
Variational formulation
.......................... 234
8.2.2
A general result
............................... 236
8.2.3
Applications
................................ 241
8.3
Existence and uniqueness in the hyperbolic case
................. 246
8.3.1
Variational formulation
.......................... 246
8.3.2
A general result
............................... 247
8.3.3
Applications
................................ 249
8.4
Qualitative properties in the parabolic case
................... 253
8.4.1
Asymptotic behaviour
........................... 253
8.4.2
The maximum principle
.......................... 255
8.4.3
Propagation at infinite velocity
...................... 256
8.4.4
Regularity and regularizing effect
..................... 257
8.4.5
Heat equation in the entire space
..................... 259
8.5
Qualitative properties in the hyperbolic case
................... 261
8.5.1
Reversibility in time
............................ 261
viii CONTENTS
8.5.2
Asymptotic behaviour and equipartition of energy
........... 262
8.5.3
Finite velocity of propagation
....................... 263
8.6
Numerical methods in the parabolic case
..................... 264
8.6.1
Semidiscretization in space
........................ 264
8.6.2
Total discretization in space-time
..................... 266
8.7
Numerical methods in the hyperbolic case
.................... 269
8.7.1
Semidiscretization in space
........................ 270
8.7.2
Total discretization in space-time
..................... 271
9
Introduction to optimization
277
9.1
Motivation and examples
............................. 277
9.1.1
Introduction
................................ 277
9.1.2
Examples
.................................. 278
9.1.3
Definitions and notation
.......................... 284
9.1.4
Optimization in finite dimensions
..................... 285
9.2
Existence of a minimum in infinite dimensions
.................. 287
9.2.1
Examples of nonexistence
......................... 287
9.2.2
Convex analysis
............................... 289
9.2.3
Existence results
.............................. 292
10
Optimality conditions and algorithms
297
10.1
Generalities
..................................... 297
10.1.1
Introduction
................................ 297
10.1.2
Differentiability
............................... 298
10.2
Optimality conditions
............................... 303
10.2.1
Euler
inequalities and convex constraints
................ 303
10.2.2 Lagrange
multipliers
............................ 306
10.3
Saddle point, Kuhn-Tucker theorem, duality
.................. 317
10.3.1
Saddle point
................................ 317
10.3.2
The Kuhn-Tucker theorem
........................ 318
10.3.3
Duality
................................... 320
10.4
Applications
..................................... 323
10.4.1
Dual or complementary energy
...................... 323
10.4.2
Optimal command
............................. 326
10.4.3
Optimization of distributed systems
................... 330
10.5
Numerical algorithms
............................... 332
10.5.1
Introduction
................................ 332
10.5.2
Gradient algorithms (case without constraints)
............. 333
10.5.3
Gradient algorithms (case with constraints)
............... 336
10.5.4
Newtoiťs
method
.............................. 342
11
Methods of operational research
(Written in collaboration with
Stephane Gaubert) 347
11.1
Introduction
..................................... 347
11.2
Linear programming
................................ 348
11.2.1
Definitions and properties
......................... 348
11.2.2
The simplex algorithm
........................... 353
11.2.3
Interior point algorithms
.......................... 357
CONTENTS ix
11.2.4
Duality
................................... 358
11.3 Integer
polyhedra
.................................. 361
11.3.1 Extreme
points of compact convex sets..................
362
11.3.2
Totally unimodular matrices
....................... 364
11.3.3
Flow problems
............................... 368
11.4
Dynamic programming
............................... 371
11.4.1
Bellman s optimality principle
....................... 372
11.4.2
Finite horizon problem
........................... 372
11.4.3
Minimum cost path, or optimal stopping, problem
........... 375
11.5
Greedy algorithms
................................. 380
11.5.1
General points about greedy methods
.................. 380
11.5.2
Kruskal s algorithm for the minimum
spanning tree problem
........................... 381
11.6
Separation and relaxation
............................. 383
11.6.1
Separation and evaluation (branch and bound)
............. 383
11.6.2
Relaxation of combinatorial problems
.................. 388
12
Appendix Review of hilbert spaces
399
13
Appendix Matrix Numerical Analysis
405
13.1
Solution of linear systems
............................. 405
13.1.1
Review of matrix norms
.......................... 406
13.1.2
Conditioning and stability
......................... 409
13.1.3
Direct methods
............................... 411
13.1.4
Iterative methods
.............................. 424
13.1.5
The conjugate gradient method
...................... 428
13.2
Calculation of eigenvalues and eigenvectors
................... 435
13.2.1
The power method
............................. 436
13.2.2
The Givens-Householder method
..................... 438
13.2.3
The Lanczos method
............................ 442
Index
451
Index notations
455
|
| any_adam_object | 1 |
| author | Allaire, Grégoire |
| author_GND | (DE-588)123414849 |
| author_facet | Allaire, Grégoire |
| author_role | aut |
| author_sort | Allaire, Grégoire |
| author_variant | g a ga |
| building | Verbundindex |
| bvnumber | BV035319444 |
| callnumber-first | Q - Science |
| callnumber-label | QA297 |
| callnumber-raw | QA297 |
| callnumber-search | QA297 |
| callnumber-sort | QA 3297 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 900 |
| ctrlnum | (OCoLC)82671667 (DE-599)BVBBV035319444 |
| dewey-full | 511.8 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 511 - General principles of mathematics |
| dewey-raw | 511.8 |
| dewey-search | 511.8 |
| dewey-sort | 3511.8 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 1. publ. |
| format | Book |
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| genre | (DE-588)4123623-3 Lehrbuch gnd-content |
| genre_facet | Lehrbuch |
| id | DE-604.BV035319444 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T21:31:12Z |
| institution | BVB |
| isbn | 0199205213 0199205221 9780199205219 9780199205226 |
| language | English French |
| lccn | 2007299208 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-017124040 |
| oclc_num | 82671667 |
| open_access_boolean | |
| owner | DE-703 DE-634 DE-20 |
| owner_facet | DE-703 DE-634 DE-20 |
| physical | xvi, 455 p. ill. 24 cm |
| publishDate | 2007 |
| publishDateSearch | 2007 |
| publishDateSort | 2007 |
| publisher | Oxford University Press |
| record_format | marc |
| series2 | Numerical mathematics and scientific computation |
| spelling | Allaire, Grégoire Verfasser (DE-588)123414849 aut Analyse numérique et optimisation Numerical analysis and optimization an introduction to mathematical modelling and numerical simulation Grégoire Allaire ; translated by Alan Craig 1. publ. Oxford Oxford University Press 2007 xvi, 455 p. ill. 24 cm txt rdacontent n rdamedia nc rdacarrier Numerical mathematics and scientific computation Includes bibliographical references (p. 447-449) and index Translated from the French. Mathematisches Modell Numerical analysis Mathematical optimization Mathematical models Numerische Mathematik (DE-588)4042805-9 gnd rswk-swf Modellierung (DE-588)4170297-9 gnd rswk-swf Finite-Elemente-Methode (DE-588)4017233-8 gnd rswk-swf Mathematisches Modell (DE-588)4114528-8 gnd rswk-swf Mehrkriterielle Optimierung (DE-588)4610682-0 gnd rswk-swf Simulation (DE-588)4055072-2 gnd rswk-swf Numerische Steuerung (DE-588)4115421-6 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Numerische Steuerung (DE-588)4115421-6 s Mehrkriterielle Optimierung (DE-588)4610682-0 s Finite-Elemente-Methode (DE-588)4017233-8 s Modellierung (DE-588)4170297-9 s DE-604 Mathematisches Modell (DE-588)4114528-8 s Simulation (DE-588)4055072-2 s Numerische Mathematik (DE-588)4042805-9 s 1\p DE-604 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017124040&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Allaire, Grégoire Numerical analysis and optimization an introduction to mathematical modelling and numerical simulation Mathematisches Modell Numerical analysis Mathematical optimization Mathematical models Numerische Mathematik (DE-588)4042805-9 gnd Modellierung (DE-588)4170297-9 gnd Finite-Elemente-Methode (DE-588)4017233-8 gnd Mathematisches Modell (DE-588)4114528-8 gnd Mehrkriterielle Optimierung (DE-588)4610682-0 gnd Simulation (DE-588)4055072-2 gnd Numerische Steuerung (DE-588)4115421-6 gnd |
| subject_GND | (DE-588)4042805-9 (DE-588)4170297-9 (DE-588)4017233-8 (DE-588)4114528-8 (DE-588)4610682-0 (DE-588)4055072-2 (DE-588)4115421-6 (DE-588)4123623-3 |
| title | Numerical analysis and optimization an introduction to mathematical modelling and numerical simulation |
| title_alt | Analyse numérique et optimisation |
| title_auth | Numerical analysis and optimization an introduction to mathematical modelling and numerical simulation |
| title_exact_search | Numerical analysis and optimization an introduction to mathematical modelling and numerical simulation |
| title_full | Numerical analysis and optimization an introduction to mathematical modelling and numerical simulation Grégoire Allaire ; translated by Alan Craig |
| title_fullStr | Numerical analysis and optimization an introduction to mathematical modelling and numerical simulation Grégoire Allaire ; translated by Alan Craig |
| title_full_unstemmed | Numerical analysis and optimization an introduction to mathematical modelling and numerical simulation Grégoire Allaire ; translated by Alan Craig |
| title_short | Numerical analysis and optimization |
| title_sort | numerical analysis and optimization an introduction to mathematical modelling and numerical simulation |
| title_sub | an introduction to mathematical modelling and numerical simulation |
| topic | Mathematisches Modell Numerical analysis Mathematical optimization Mathematical models Numerische Mathematik (DE-588)4042805-9 gnd Modellierung (DE-588)4170297-9 gnd Finite-Elemente-Methode (DE-588)4017233-8 gnd Mathematisches Modell (DE-588)4114528-8 gnd Mehrkriterielle Optimierung (DE-588)4610682-0 gnd Simulation (DE-588)4055072-2 gnd Numerische Steuerung (DE-588)4115421-6 gnd |
| topic_facet | Mathematisches Modell Numerical analysis Mathematical optimization Mathematical models Numerische Mathematik Modellierung Finite-Elemente-Methode Mehrkriterielle Optimierung Simulation Numerische Steuerung Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017124040&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT allairegregoire analysenumeriqueetoptimisation AT allairegregoire numericalanalysisandoptimizationanintroductiontomathematicalmodellingandnumericalsimulation |