  We consider the use of Gaussian process (GP) priors for solving inverse
problems in a Bayesian framework. As is well known, the computational
complexity of GPs scales cubically in the number of datapoints. We here show
that in the context of inverse problems involving integral operators, one faces
additional difficulties that hinder inversion on large grids. Furthermore, in
that context, covariance matrices can become too large to be stored. By
leveraging results about sequential disintegrations of Gaussian measures, we
are able to introduce an implicit representation of posterior covariance
matrices that reduces the memory footprint by only storing low rank
intermediate matrices, while allowing individual elements to be accessed
on-the-fly without needing to build full posterior covariance matrices.
Moreover, it allows for fast sequential inclusion of new observations. These
features are crucial when considering sequential experimental design tasks. We
demonstrate our approach by computing sequential data collection plans for
excursion set recovery for a gravimetric inverse problem, where the goal is to
provide fine resolution estimates of high density regions inside the Stromboli
volcano, Italy. Sequential data collection plans are computed by extending the
weighted integrated variance reduction (wIVR) criterion to inverse problems.
Our results show that this criterion is able to significantly reduce the
uncertainty on the excursion volume, reaching close to minimal levels of
residual uncertainty. Overall, our techniques allow the advantages of
probabilistic models to be brought to bear on large-scale inverse problems
arising in the natural sciences.
