Numerical Data Fitting in Dynamical Systems

Real 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.

Klappentext

The main objective of the book is to give an overview of numerical methods to compute parameters of a dynamical model by a least squares fit of experimental data. The mathematical equations under consideration are explicit model functions or steady state systems in the simplest case, or responses of dynamical systems defined by ordinary differential equations, differential algebraic equations, partial differential equations, and partial differential algebraic equations (1D). Many different mathematical disciplines must be combined to find a solution, for example nonlinear programming, least squares optimization, systems of nonlinear equations, ordinary differential equations, discretization of partial differential equations, sensitivity analysis, automatic differentiation, and statistics.

Inhalt
1 Introduction.- 2 Mathematical Foundations.- 3 Data Fitting Models.- 4 Numerical Experiments.- 5 Case Studies.- Appendix A: Software Installation.- 1 Hardware and Software Requirements.- 2 System Setup.- 3 Packing List.- Appendix B: Test Examples.- 1 Explicit Model Functions.- 2 Laplace Transforms.- 3 Steady State Equations.- 4 Ordinary Differential Equations.- 5 Differential Algebraic Equations.- 6 Partial Differential Equations.- 7 Partial Differential Algebraic Equations.- Appendix C: The PCOMP Language.- Appendix D: Generation of Fortran Code.- 1 Model Equations.- 1.1 Input of Explicit Model Functions.- 1.2 Input of Laplace Transformations.- 1.3 Input of Systems of Steady State Equations.- 1.4 Input of Ordinary Differential Equations.- 1.5 Input of Differential Algebraic Equations.- 1.6 Input of Time-Dependent Partial Differential Equations.- 1.7 Input of Partial Differential Algebraic Equations.- 2 Execution of Generated Code.- References.