The Nature of Mathematical ModelingCambridge University Press, 1999 - 344 lappuses This book first covers exact and approximate analytical techniques (ordinary differential and difference equations, partial differential equations, variational principles, stochastic processes); numerical methods (finite differences for ODE's and PDE's, finite elements, cellular automata); model inference based on observations (function fitting, data transforms, network architectures, search techniques, density estimation); as well as the special role of time in modeling (filtering and state estimation, hidden Markov processes, linear and nonlinear time series). Each of the topics in the book would be the worthy subject of 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. |
Saturs
Introduction | 1 |
Ordinary Differential and Difference Equations 259 | 9 |
Variational Principles | 34 |
Random Systems 271 | 44 |
Random Systems | 46 |
13 | 47 |
9 | 60 |
Ordinary Differential Equations | 67 |
Architectures 309 | 138 |
Architectures | 139 |
43 | 153 |
Optimization and Search | 156 |
Clustering and Density Estimation | 169 |
Filtering and State Estimation | 186 |
34 | 193 |
48 | 196 |
Partial Differential Equations | 78 |
Finite Elements | 95 |
Cellular Automata and Lattice Gases 292 | 102 |
Cellular Automata and Lattice Gases | 104 |
Function Fitting 302 | 113 |
Function Fitting | 115 |
Transforms | 128 |
67 | 202 |
Linear and Nonlinear Time Series | 204 |
102 | 214 |
Graphical and Mathematical Software | 225 |
Problems | 249 |
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algorithm apply approximation assume basis functions boundary conditions calculation called cellular automata chapter cluster cluster-weighted coefficients complex covariance matrix data set defined degrees of freedom density estimation depends derivative diagonal dimension discrete embedding entropy error example expansion finite elements Fourier transform function evaluations Gaussian Gershenfeld given GLfloat global gradient descent important initial conditions input integral internal inverse iterative Kalman filtering Laplace transform lattice least squares linear log₂ mean measurements method minimization minimum multiplying needed Neil Gershenfeld neural networks noise nonlinear observable optimal orthogonal orthonormal output p(cm p(xt Padé approximant parameters partial differential equations particle polynomial possible predict probability distribution problem random number random variables requires signal simple simulated annealing singular values solution solved space step stochastic techniques update vanish variance vector wavelets Wiener filters z-transform zero