In least-squares problems the target function of the optimization problem is built from the square of the two-norm (see Appendix B.3) of the residual vector. This residual is usually computed by comparing the measured and calculated data.
The model
in Figure 4.1 determines a set of simulated data
from a parameter vector
which is
compared with measured data. The model can consist of one or several
simulation steps with additional data conversions and result extraction
steps.
The parameter vector
contains various simulation parameters, used
during the evaluation of model
.
The resulting data vector
is the extracted result of the
performed simulations. The residuum vector
is calculated by
subtracting the measured data
from the result vector
.
In general the model
constitutes a non-linear function and
the optimization problem is of a nonlinear least-squares type4.3.
Normally the extracted data vector
is dependent on the
available measured data. So the simulations have to be performed at
the same data points as the measurements, or general results have to
be interpolated by the tool calculating the residuals. These
dependencies are not shown in
Figure 4.1.
Common nonlinear least-squares optimization problems occur in data
fit, calibration, and inverse modeling tasks.