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.