  Inferring interior properties of the Sun from photospheric measurements of
the seismic wavefield constitutes the helioseismic inverse problem. Deviations
in seismic measurements (such as wave travel times) from their fiducial values
estimated for a given model of the solar interior imply that the model is
inaccurate. Contemporary inversions in local helioseismology assume that
properties of the solar interior are linearly related to measured travel-time
deviations. It is widely known, however, that this assumption is invalid for
sunspots and active regions, and likely for supergranular flows as well. Here,
we introduce nonlinear optimization, executed iteratively, as a means of
inverting for the sub-surface structure of large-amplitude perturbations.
Defining the penalty functional as the $L_2$ norm of wave travel-time
deviations, we compute the the total misfit gradient of this functional with
respect to the relevant model parameters %(only sound speed in this case) at
each iteration around the corresponding model. The model is successively
improved using either steepest descent, conjugate gradient, or quasi-Newton
limited-memory BFGS. Performing nonlinear iterations requires privileging
pixels (such as those in the near-field of the scatterer), a practice not
compliant with the standard assumption of translational invariance.
Measurements for these inversions, although similar in principle to those used
in time-distance helioseismology, require some retooling. For the sake of
simplicity in illustrating the method, we consider a 2-D inverse problem with
only a sound-speed perturbation.
