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.
Crestel, B., Stadler, G., & Ghattas, O. (2018). A comparative study of structural similarity and regularization for joint inverse problems governed by PDEs. Inverse Problems, 35(2), 024003. doi:10.1088/1361-6420/aaf129
The authors would like to thank David Keyes (KAUST) and George Turkiyyah (AUB) for very helpful discussions that inspired this work. They would also like to thank Sergey Fomel (UT-Austin) for referring them to [2, 3], Jan Modersitzki (Lübeck) for directing their attention to
, Nick Alger (UT-Austin) for help with figure 1, and two anonymous referees whose comments helped significantly to improve the manuscript. This work was partially supported by AFOSR grant FA9550-17-1-0190, DOE grant DE-SC0009286, KAUST award OSR-2016-CCF-2596, and NSF grants ACI-1550593, DMS-1723211, CBET-1507009, and CBET-1508713.