The nature of mathematical modeling:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge [u.a.]
Cambridge Univ. Press
2011
|
| Edition: | 1. paperback ed. |
| Subjects: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025217985&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025217985&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| Physical Description: | XII, 344 S. Ill., graph. Darst. |
| ISBN: | 052121050X 9780521210508 0521570956 |
Staff View
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| 245 | 1 | 0 | |a The nature of mathematical modeling |c Neil Gershenfeld |
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| 264 | 1 | |a Cambridge [u.a.] |b Cambridge Univ. Press |c 2011 | |
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Record in the Search Index
| _version_ | 1819275659659182080 |
|---|---|
| adam_text | .
„ack
Re-issue
This book, about the nature and techniques of mathematical modeling,
is oriented towards simple efficient implementations on computers.
The text is in three sections. The first covers exact and approximate
analytical techniques; the second, numerical methods; the third, model
inference based on observations; and the last, the special role of time
in modeling.
Each of the topics in the book would be the worthy subject ot a
dedicated text, but only by presenting the material in this way is it
possible to make so much material accessible to so many people. Each
chapter presents a concise summary of the core results in an area,
providing an orientation to what they can (and cannot) do, enough
background to use them to solve typical problems, and pointers to access
the literature for particular applications. The text is complemented by
extensive worked problems.
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Contents
page
χ
Preface
xi
1
Introduction
1
1.1
Selected References
.......................... 3
Part One: Analytical Models
5
2
Ordinary Differential and Difference Equations
9
2.1
Linear Differential Equations
..................... 9
2.2
Systems of Differential Equations and Normal Modes
......... 12
2.3
Laplace Transforms
.......................... 13
2.4
Perturbation Expansions
........................ 17
2.5
Discrete Time Equations
........................ 18
2.6
z-Transforms
.............................. 19
2.7
Selected References
.......................... 21
2.8
Problems
................................ 22
3
Partial Differential Equations
24
3.1
The Origin of Partial Differential Equations
.............. 24
3.2
Linear Partial Differential Equations
.................. 26
3.3
Separation of Variables
......................... 27
3.3.1
Rectangular Coordinates
.................... 28
3.3.2
Cylindrical Coordinates
.................... 29
3.3.3
Spherical Coordinates
..................... 31
3.4
Transform Techniques
......................... 32
3.5
Selected References
.......................... 33
3.6
Problems
................................ 33
4
Variational Principles
34
4.1
Variational Calculus
.......................... 34
4.1.1
Euler s Equation
........................ 34
4.1.2
Integrals and Missing Variables
................ 36
4.1.3
Constraints and Lagrange Multipliers
............. 37
vi
Contents
4.2
Varíational
Problems.......................... 38
4.2.1
Optics:
Fermaťs
Principle
................... 38
4.2.2
Analytical Mechanics: Hamilton s Principle
.......... 38
4.2.3
Symmetry: Noether s Theorem
................ 39
4.3
Rigid Body Motion
........................... 40
4.4
Selected References
.......................... 43
4.5
Problems
................................ 43
5
Random Systems
44
5.1
Random Variables
........................... 44
5.1.1
Joint Distributions
....................... 46
5.1.2
Characteristic Functions
.................... 48
5.2
Stochastic Processes
.......................... 50
5.2.1
Distribution Evolution Equations
............... 51
5.2.2
Stochastic Differential Equations
............... 55
5.3
Random Number Generators
...................... 56
5.3.1
Linear Congruential
...................... 57
5.3.2
Linear Feedback
........................ 59
5.4
Selected References
.......................... 60
5.5
Problems
................................ 60
Part Two: Numerical Models
63
6
Finite Differences: Ordinary Differential Equations
67
6.1
Numerical Approximations
....................... 67
6.2
Runge-Kutta Methods
......................... 70
6.3
Beyond Runge-Kutta
......................... 72
6.4
Selected References
.......................... 76
6.5
Problems
................................ 76
7
Finite Differences: Partial Differential Equations
78
7.1
Hyperbolic Equations: Waves
..................... 79
7.2
Parabolic Equations: Diffusion
..................... 81
7.3
Elliptic Equations: Boundary Values
.................. 84
7.4
Selected References
.......................... 91
7.5
Problems
................................ 91
8
Finite Elements
93
8.1
Weighted Residuals
.......................... 93
8.2
Rayleigh-Ritz
Varíational
Methods
................... 99
8.3
Selected References
.......................... 100
8.4
Problems
................................ 101
9
Cellular Automata and Lattice Gases
102
9.1
Lattice Gases and Fluids
........................ 103
9.2
Cellular Automata and Computing
................... 107
Contents
vii
9.3 Selected
References
.......................... 109
9.4 Problems................................ HO
Part Three: Observational
Models
111
10
Function Fitting
115
10.1
Model Estimation
........................... 116
10.2
Least Squares
............................. 117
10.3
Linear Least Squares
.......................... 118
Ю.
3.1
Singular Value Decomposition
................. 119
10.4
Nonlinear Least Squares
........................ 122
10.4.1
Levenberg-Marquardt Method
................ 124
10.5
Estimation, Fisher Information, and the Cramer—Rao Inequality
.... 125
10.6
Selected References
.......................... 127
10.7
Problems
................................ 127
11
Transforms
128
11.1
Orthogonal Transforms
........................ 128
11.2
Fourier Transforms
.......................... 129
11.3
Wavelets
................................ 131
11.4
Principal Components
......................... 136
11.5
Selected References
.......................... 138
11.6
Problems
................................ 138
12
Architectures
139
12.1
Polynomials
.............................. 139
12.1.1
Padé Approximants
...................... 139
12.1.2
Splines
............................. 141
12.2
Orthogonal Functions
......................... 142
12.3
Radial Basis Functions
......................... 145
12.4
Overfitting
............................... 147
12.5
Curse of Dimensionality
........................ 148
12.6
Neural Networks
............................ 150
12.6.1
Back Propagation
........................ 152
12.7
Regularization
............................. 153
12.8
Selected References
.......................... 155
12.9
Problems
................................ 155
13
Optimization and Search
156
13.1
Multidimensional Search
........................ 157
13.2
Local Minima
............................. 161
13.3
Simulated Annealing
.......................... 162
13.4
Genetic Algorithms
........................... 164
13.5
The Blessing of Dimensionality
.................... 166
viii Contents
13.6
Selected References
.......................... 167
13.7
Problems
................................ 168
14
Clustering and Density Estimation
169
14.1
Histogramming, Sorting, and Trees
.................. 169
14.2
Fitting Densities
............................ 172
14.3
Mixture Density Estimation and Expectation-Maximization
...... 174
14.4
Cluster-Weighted Modeling
...................... 178
14.5
Selected References
.......................... 185
14.6
Problems
................................ 185
15
Filtering and State Estimation
186
15.1
Matched Filters
............................ 186
15.2
Wiener Filters
............................. 187
15.3
Kalman
Filters
............................. 189
15.4
Nonlinearity and Entrainment
..................... 195
15.5
Hidden Markov Models
........................ 197
15.6
Selected References
.......................... 203
15.7
Problems
................................ 203
16
Linear and Nonlinear Time Series
204
16.1
Linear Time Series
........................... 205
16.2
The Breakdown of Linear Systems Theory
.............. 207
16.3
State-Space Reconstruction
...................... 208
16.4
Characterization
............................ 213
16.4.1
Dimensions
.......................... 214
16.4.2
Lyapunov Exponents
...................... 216
16.4.3
Entropies
............................ 217
16.5
Forecasting
............................... 220
16.6
Selected References
.......................... 224
16.7
Problems
................................ 224
Appendix
1
Graphical and Mathematical Software
225
ALI Math
Packages
............................. 226
Al.
1.1
Programming Environments
.................. 226
Al.
1.2
Interactive Environments
................... 228
A1.2 Graphics Tools
............................. 230
Al.2.1 Postscript
........................... 230
Al.2.2 X Windows
.......................... 234
Al.2.3
OpenGL............................ 240
Al.2.4 Java
.............................. 244
A1.3 Problems
................................ 249
Appendix
2
Network Programming
250
A2.1
OSI,
TCP/IP, and All That
...................... 250
Contents ix
A2.2 Socket I/O............................... 251
A2.3 Parallel Programming ......................... 254
Appendix 3
Benchmarking
257
Appendix 4 Problem Solutions 259
A4.1
Introduction
.............................. 259
A4.2
Ordinary
Differential
and Difference Equations............
259
A4.3
Partial
Differential
Equations
...................... 266
A4.4 Variational
Principles
.......................... 269
A4.5
Random Systems
............................ 271
A4.6 Finite Differences: Ordinary Differential Equations
.......... 276
A4.7 Finite Differences: Partial Differential Equations
........... 281
A4.8 Finite Elements
............................ 289
A4.9 Cellular Automata and Lattice Gases
.................. 292
A4.10 Function Fitting
............................ 302
A4.11 Transforms
............................... 305
A4.12
Architectures
.............................. 309
A4.13
Optimization and Search
........................ 315
A4.14 Clustering and Density Estimation
................... 319
A4.15
Filtering and State Estimation
..................... 323
A4.16 Linear and Nonlinear Time Series
................... 325
Bibliography
330
Index
340
|
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| author | Gershenfeld, Neil A. 1959- |
| author_GND | (DE-588)173281656 |
| author_facet | Gershenfeld, Neil A. 1959- |
| author_role | aut |
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| classification_rvk | SK 110 |
| ctrlnum | (OCoLC)746289115 (DE-599)BVBBV040364221 |
| 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 18 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 1. paperback ed. |
| format | Book |
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| id | DE-604.BV040364221 |
| illustrated | Illustrated |
| indexdate | 2024-12-20T16:13:29Z |
| institution | BVB |
| isbn | 052121050X 9780521210508 0521570956 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-025217985 |
| oclc_num | 746289115 |
| open_access_boolean | |
| owner | DE-20 DE-19 DE-BY-UBM DE-703 DE-1102 DE-523 |
| owner_facet | DE-20 DE-19 DE-BY-UBM DE-703 DE-1102 DE-523 |
| physical | XII, 344 S. Ill., graph. Darst. |
| publishDate | 2011 |
| publishDateSearch | 2011 |
| publishDateSort | 2011 |
| publisher | Cambridge Univ. Press |
| record_format | marc |
| spellingShingle | Gershenfeld, Neil A. 1959- The nature of mathematical modeling Mathematisches Modell (DE-588)4114528-8 gnd Computersimulation (DE-588)4148259-1 gnd |
| subject_GND | (DE-588)4114528-8 (DE-588)4148259-1 |
| title | The nature of mathematical modeling |
| title_auth | The nature of mathematical modeling |
| title_exact_search | The nature of mathematical modeling |
| title_full | The nature of mathematical modeling Neil Gershenfeld |
| title_fullStr | The nature of mathematical modeling Neil Gershenfeld |
| title_full_unstemmed | The nature of mathematical modeling Neil Gershenfeld |
| title_short | The nature of mathematical modeling |
| title_sort | the nature of mathematical modeling |
| topic | Mathematisches Modell (DE-588)4114528-8 gnd Computersimulation (DE-588)4148259-1 gnd |
| topic_facet | Mathematisches Modell Computersimulation |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025217985&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025217985&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
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