  We consider a class of inverse problems where it is possible to aggregate the
results of multiple experiments. This class includes problems where the forward
model is the solution operator to linear ODEs or PDEs. The tremendous size of
such problems motivates dimensionality reduction techniques based on randomly
mixing experiments. These techniques break down, however, when robust
data-fitting formulations are used, which are essential in cases of missing
data, unusually large errors, and systematic features in the data unexplained
by the forward model. We survey robust methods within a statistical framework,
and propose a semistochastic optimization approach that allows dimensionality
reduction. The efficacy of the methods are demonstrated for a large-scale
seismic inverse problem using the robust Student's t-distribution, where a
useful synthetic velocity model is recovered in the extreme scenario of 60%
data missing at random. The semistochastic approach achieves this recovery
using 20% of the effort required by a direct robust approach.
