  Joint inversion refers to the simultaneous inference of multiple parameter
fields from observations of systems governed by single or multiple forward
models. In many cases these parameter fields reflect different attributes of a
single medium and are thus spatially correlated or structurally similar. By
imposing prior information on their spatial correlations via a joint
regularization term, we seek to improve the reconstruction of the parameter
fields relative to inversion for each field independently. One of the main
challenges is to devise a joint regularization functional that conveys the
spatial correlations or structural similarity between the fields while at the
same time permitting scalable and efficient solvers for the joint inverse
problem. We describe several joint regularizations that are motivated by these
goals: a cross-gradient and a normalized cross-gradient structural similarity
term, the vectorial total variation, and a joint regularization based on the
nuclear norm of the gradients. Based on numerical results from three classes of
inverse problems with piecewise-homogeneous parameter fields, we conclude that
the vectorial total variation functional is preferable to the other methods
considered. Besides resulting in good reconstructions in all experiments, it
allows for scalable, efficient solvers for joint inverse problems governed by
PDE forward models.
