### 6.4 Optimization approach

The optimization approach as proposed in Wheatland et al. (2000) is closely related to the MHD
relaxation approach. It shares with this method that a similar initial non-equilibrium state is
iterated towards a NLFFF equilibrium. It solves a similar iterative equation as Equation (65)
but has additional terms, as explained below. The force-free and solenoidal conditions are solved by
minimizing the functional
If the minimum of this functional at is attained then the NLFFF equations (2) – (4) are fulfilled.
The functional is minimized by taking the functional derivative of Equation (68) with respect to an
iteration parameter :
For vanishing surface terms the functional decreases monotonically if the magnetic field is iterated by
The first term in Equation (70) is identical with as defined in Equation (66).
A principal problem with the optimization and the MHD-relaxation approaches is that using the full
magnetic field vector on the lower boundary does not guarantee the existence of a force-free configuration
(see the consistency criteria in Section 4.6. Consequently, if fed with inconsistent boundary data,
the codes cannot find a force-free configuration, but a finite residual Lorentz force and/or a
finite divergence of the field remains in the 3D equilibrium. A way around this problem is to
preprocess the measured photospheric data, as explained in Section 4.7. An alternative approach
is that one allows deviations of the measured horizontal field vector and the corresponding
field vector on the lower boundary of the computational box during the minimization of the
functional (68). Wiegelmann and Inhester (2010) extended this functional by another term

where is a free parameter and the matrix contains information how reliable the data (mainly
measurements of the horizontal photospheric field) are. With this approach inconsistencies in the
measurement lead to a solution compatible with physical requirements (vanishing Lorentz force and
divergence), leaving differences between and the bottom boundary field in regions where is
low (and the measurement error high). Consequently, this approach takes measurement errors, missing data,
and data inconsistencies into account. Further tests are necessary to investigate whether this approach or
preprocessing, or a combination of both, is the most effective way to deal with noisy and inconsistent
photospheric field measurements. This approach, as well as a variant of the Grad–Rubin method, have been
developed in response to a joint study by DeRosa et al. (2009), where one of the main findings w as that
force-free extrapolation codes should be able to incorporate measurement inconsistencies (see also
Section 6.6).