The Nature of Mathematical Modeling

Cover
Title page
Preface
1 Introduction
1.1 Selected References
Part One: Analytical Models
2 Ordinary Differential and Difference Equations
2.1 Linear Differentiai Equations
2.2 Systems of Differentiai Equations and Normal Modes
2.3 Laplace Transforms
2.4 Perturbation Expansions
2.5 Discrete Time Equations
2.6 z-Transforms
2.7 Selected References
2.8 Problems
3 Partial Differential Equations
3.1 The Origin of Partial Differentiai Equations
3.2 Linear Partial Differentiai Equations
3.3 Separation of Variables
3.3.1 Rectangular Coordinates
3.3.2 Cylindrical Coordinates
3.3.3 Spherical Coordinates
3.4 Transform Techniques
3.5 Selected References
3.6 Problems
4 Variational Principles
4.1 Variational Calculus
4.1.1 Euler's Equation
4.1.2 Integrals and Missing Variables
4.1.3 Constraints and Lagrange Multipliers
4.2 Variational Problems
4.2.1 Optics: Fermat's Principle
4.2.2 Analytical Mechanics: Hamilton's Principle
4.2.3 Symmetry; Noether's Theorem
4.3 Rigid Body Motion
4.4 Selected References
4.5 Problems
5 Random Systems
5.1 Random Variables
5.1.1 Joint Distributions
5.1.2 Characteristic Functions
5.2 Stochastic Processes
5.2.1 Distribution Evolution Equations
5.2.2 Stochastic Differentiai Equations
5.3 Random Number Generators
5.3.1 Linear Congruential
5.3.2 Linear Feedback
5.4 Selected References
5.5 Problems
Part Two: Numerical Models
6 Finite Differences: Ordinary Differential Equations
6.1 Numerical Approximations
6.2 Runge-Kutta Methods
6.3 Beyond Runge-Kutta
6.4 Selected References
6.5 Problems
7 Finite Differences: Partial Differential Equations
7.1 Hyperbolic Equations: Waves
7.2 Parabolic Equations: Diffusion
7.3 Elliptic Equations: Boundary Values
7.4 Selected References
7.5 Problems
8 Finite Elements
8.1 Weighted Residuals
8.2 Rayleigh-Ritz Variational Methods
8.3 Selected References
8.4 Problems
9 Cellular Automata and Lattice Gases
9.1 Lattice Gases and Fluids
9.2 Cellular Automata and Computing
9.3 Selected References
9.4 Problems
Part Three: Observational Models
10 Function Fitting
10.1 Model Estimation
10.2 Least Squares
10.3 Linear Least Squares
10.3.1 Singular Value Decomposition
10.4 Nonlinear Least Squares
10.4.1 Levenberg-Marquardt Method
10.5 Estimation, Fisher Information, and the Cramér-Rao Inequality
10.6 Selected References
10.7 Problems
11 Transforms
11.1 Orthogonal Transforms
11.2 Fourier Transforms
11.3 Wavelets
11.4 Principal Components
11.5 Selected References
11.6 Problems
12 Architectures
12.1 Polynomials
12.1.1 Padé Approximants
12.1.2 Splines
12.2 Orthogonal Functions
12.3 Radial Basis Functions
12.4 Overfitting
12.5 Curse of Dimensionality
12.6 Neural Networks
12.6.1 Back Propagation
12.7 Regularization
12.8 Selected References
12.9 Problems
13 Optimization and Search
13.1 Multidimensional Search
13.2 Local Minima
13.3 Simulated Annealing
13.4 Genetic Algorithms
13.5 The Blessing of Dimensionality
13.6 Selected References
13.7 Problems
14 Clustering and Density Estimation
14.1 Histogramming, Sorting, and Trees
14.2 Fitting Densities
14.3 Mixture Density Estimation and Expectation-Maximization
14.4 Cluster-Weighted Modeling
14.5 Selected References
14.6 Problems
15 Filtering and State Estimation
15.1 Matched Filters
15.2 Wiener Filters
15.3 Kalman Filters
15.4 Nonlinearity and Entrainment
15.5 Hidden Markov Models
15.6 Selected References
15.7 Problems
16 Linear and Nonlinear Time Series
16.1 Linear Time Series
16.2 The Breakdown of Linear Systems Theory
16.3 State-Space Reconstruction
16.4 Characterization
16.4.1 Dimensions
16.4.2 Lyapunov Exponents
16.4.3 Entropies
16.5 Forecasting
16.6 Selected References
16.7 Problems
Appendix 1 Graphical and Mathematical Software
A1.1 Math Packages
A1.1.1 Programming Environments
Al.l.2 Interactive Environments
Al.2 Graphics Tools
A1.2.1 Postscript
Al.2.2 X Windows
Al.2.3 OpenGL
Al.2.4 Java
Al.3 Problems
Appendix 2 Network Programming
A2.1 OSI, TCP/IP, and All That
A2.2 Socket 1/0
A2.3 Parallel Programming
Appendix 3 Benchmarking
Appendix 4 Problem Solutions
A4.1 Introduction
A4.2 Ordinary Differentiai and Difference Equations
A4.3 Partial Differentiai Equations
A4.4 Variational Principles
A4.5 Random Systems
A4.6 Finite Differences: Ordinary Differentiai Equations
A4.7 Finite Differences: Partial Differentiai Equations
A4.8 Finite Elements
A4.9 Cellular Automata and Lattice Gases
A4.10 Function Fitting
A4.11 Transforms
A4.12 Architectures
A4.13 Optimization and Search
A4.14 Clustering and Density Estimation
A4.15 Filtering and State Estimation
A4.16 Linear and Nonlinear Time Series
Bibliography
Index