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We consider the use of Gaussian process (GP) priors for solving inverseproblems in a Bayesian framework. As is well known, the computationalcomplexity of GPs scales cubically in the number of datapoints. We here showthat in the context of inverse problems involving integral operators, one facesadditional difficulties that hinder inversion on large grids. Furthermore, inthat context, covariance matrices can become too large to be stored. Byleveraging results about sequential disintegrations of Gaussian measures, weare able to introduce an implicit representation of posterior covariancematrices that reduces the memory footprint by only storing low rankintermediate matrices, while allowing individual elements to be accessedon-the-fly without needing to build full posterior covariance matrices.Moreover, it allows for fast sequential inclusion of new observations. Thesefeatures are crucial when considering sequential experimental design tasks. Wedemonstrate our approach by computing sequential data collection plans forexcursion set recovery for a gravimetric inverse problem, where the goal is toprovide fine resolution estimates of high density regions inside the Strombolivolcano, Italy. Sequential data collection plans are computed by extending theweighted integrated variance reduction (wIVR) criterion to inverse problems.Our results show that this criterion is able to significantly reduce theuncertainty on the excursion volume, reaching close to minimal levels ofresidual uncertainty. Overall, our techniques allow the advantages ofprobabilistic models to be brought to bear on large-scale inverse problemsarising in the natural sciences.

Uncertainty Quantification and Experimental Design for Large-Scale Linear Inverse Problems under Gaussian Process Priors