The Nature of Mathematical ModelingThis 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. |
Contents
Ordinary Differential and Difference Equations | 9 |
Partial Differential Equations | 24 |
Variational Principles | 34 |
Random Systems | 44 |
Ordinary Differential Equations | 67 |
Partial Differential Equations | 78 |
Finite Elements | 93 |
Cellular Automata and Lattice Gases | 102 |
Transforms | 128 |
Architectures 309 | 138 |
Architectures | 139 |
Optimization and Search | 156 |
Clustering and Density Estimation | 169 |
Filtering and State Estimation | 186 |
Linear and Nonlinear Time Series | 204 |
Graphical and Mathematical Software | 225 |
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Common terms and phrases
algorithm apply approximation basis functions boundary conditions calculation called cellular automata chapter cluster coefficients complex coordinates covariance matrix data set defined degrees of freedom density estimation depends derivative diagonal dimension discrete embedding entropy error Euler's equation example expansion Figure finite elements Fourier transform Gaussian Gershenfeld given GLfloat global gradient gradient descent important initial conditions input integral internal inverse iterative Kalman filter Laplace transform lattice linear log2 measurements method minimize needed Neil Gershenfeld noise nonlinear NPTS observable orthogonal oscillators output p(cm p(xt parameters partial differential equations particle polynomial possible PostScript predict probability distribution problem random number random variable requires signal simple simplex simulated annealing solution solved space step stochastic techniques un+1 update variance vector wavelets Wiener filter z-transform zero Δα ди ду ут นี ปี