Numerical Data Fitting in Dynamical Systems: A Practical Introduction with Applications and SoftwareReal life phenomena in engineering, natural, or medical sciences are often described by a mathematical model with the goal to analyze numerically the behaviour of the system. Advantages of mathematical models are their cheap availability, the possibility of studying extreme situations that cannot be handled by experiments, or of simulating real systems during the design phase before constructing a first prototype. Moreover, they serve to verify decisions, to avoid expensive and time consuming experimental tests, to analyze, understand, and explain the behaviour of systems, or to optimize design and production. As soon as a mathematical model contains differential dependencies from an additional parameter, typically the time, we call it a dynamical model. There are two key questions always arising in a practical environment: 1 Is the mathematical model correct? 2 How can I quantify model parameters that cannot be measured directly? In principle, both questions are easily answered as soon as some experimental data are available. The idea is to compare measured data with predicted model function values and to minimize the differences over the whole parameter space. We have to reject a model if we are unable to find a reasonably accurate fit. To summarize, parameter estimation or data fitting, respectively, is extremely important in all practical situations, where a mathematical model and corresponding experimental data are available to describe the behaviour of a dynamical system. |
Contents
INTRODUCTION | 1 |
MATHEMATICAL FOUNDATIONS | 7 |
12 Convexity and Constraint Qualification | 9 |
13 Necessary and Sufficient Optimality Criteria | 10 |
2 Sequential Quadratic Programming Methods | 14 |
22 Line Search and QuasiNewton Updates | 16 |
23 Convergence | 18 |
24 Systems of Nonlinear Equations | 20 |
55 Integration Areas and Transition Conditions | 162 |
56 Switching Points | 167 |
57 Constraints | 169 |
6 Optimal Control Problems | 175 |
NUMERICAL EXPERIMENTS | 181 |
1 Test Environment | 182 |
2 Numerical Pitfalls | 183 |
22 Slow Convergence | 186 |
3 Least Squares Methods | 23 |
32 GaussNewton and Related Methods | 24 |
33 Solution of Least Squares Problems by SQP Methods | 27 |
34 Constrained Least Squares Optimization | 31 |
35 Alternative Norms | 33 |
4 Numerical Solution of Ordinary Differential Equations | 38 |
42 Implicit Solution Methods | 40 |
43 Sensitivity Equations | 43 |
44 Internal Numerical Differentiation | 46 |
5 Numerical Solution of Differential Algebraic Equations | 48 |
52 Index of a Differential Algebraic Equation | 50 |
53 Index Reduction and Drift Effect | 52 |
54 Projection Methods | 55 |
55 Consistent Initial Values | 60 |
56 Implicit Solution Methods | 62 |
6 Numerical Solution of OneDimensional Partial Differential Equations | 66 |
62 Some Special Classes of Equations | 68 |
63 The Method of Lines | 74 |
64 Partial Differential Algebraic Equations | 78 |
65 Difference Formulae | 81 |
66 Polynomial Interpolation | 84 |
67 Upwind Formulae for Hyperbolic Equations | 85 |
68 Essentially NonOscillatory Schemes | 93 |
69 Systems of Hyperbolic Equations | 98 |
610 Sensitivity Equations | 101 |
7 Laplace Transforms | 104 |
72 Numerical BackTransformation | 107 |
8 Automatic Differentiation | 109 |
82 Reverse Mode | 112 |
9 Statistical Interpretation of Results | 115 |
DATA FITTING MODELS | 119 |
1 Explicit Model Functions | 120 |
2 Laplace Transforms | 124 |
3 Steady State Equations | 126 |
4 Ordinary Differential Equations | 128 |
42 Differential Algebraic Equations | 129 |
43 Switching Points | 131 |
44 Constraints | 137 |
45 Shooting Method | 141 |
46 Boundary Value Problems | 146 |
47 Variable Initial Times | 148 |
5 Partial Differential Equations | 151 |
52 Partial Differential Algebraic Equations | 153 |
53 Flux Functions | 154 |
54 Coupled Ordinary Differential Algebraic Equations | 157 |
23 Badly Scaled Data and Parameters | 189 |
24 NonIdentiflability of Models | 192 |
25 Errors in Experimental Data | 195 |
26 Inconsistent Constraints | 197 |
27 NonDifferentiable Model Functions | 201 |
28 Oscillating Model Functions | 205 |
3 Testing the Validity of Models | 208 |
32 Statistical Analysis | 210 |
33 Constraints | 212 |
4 Performance Evaluation | 216 |
42 Individual Numerical Results | 218 |
CASE STUDIES | 231 |
2 ReceptorLigand Binding Study | 236 |
3 Robot Design | 239 |
4 Multibody System of a Truck | 243 |
5 Binary Distillation Column | 248 |
6 Acetylene Reactor | 252 |
7 Transdermal Application of Drugs | 257 |
8 Groundwater Flow | 263 |
9 Cooling a Hot Strip Mill | 266 |
10 Drying Maltodextrin in a Convection Oven | 269 |
11 Fluid Dynamics of Hydro Systems | 273 |
12 Horn Radiators for Satellite Communication | 278 |
Software Installation | 285 |
3 Packing List | 286 |
Test Examples | 287 |
1 Explicit Model Functions | 288 |
2 Laplace Transforms | 295 |
3 Steady State Equations | 296 |
4 Ordinary Differential Equations | 299 |
5 Differential Algebraic Equations | 317 |
6 Partial Differential Equations | 320 |
7 Partial Differential Algebraic Equations | 331 |
The PCOMP Language | 335 |
Generation of Fortran Code | 345 |
12 Input of Laplace Transformations | 346 |
13 Input of Systems of Steady State Equations | 347 |
14 Input of Ordinary Differential Equations | 348 |
15 Input of Differential Algebraic Equations | 349 |
16 Input of TimeDependent Partial Differential Equations | 350 |
17 Input of Partial Differential Algebraic Equations | 352 |
2 Execution of Generated Code | 355 |
References | 359 |
387 | |
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Numerical Data Fitting in Dynamical Systems: A Practical Introduction with ... Klaus Schittkowski No preview available - 2013 |
Common terms and phrases
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