Efficient Gaussian process updating under linear operator data for uncertainty reduction on implicit sets in Bayesian inverse problems

This thesis aims at developing sequential uncertainty reduction techniques for set estimation in Bayesian inverse problems. Sequential uncertainty reduction (SUR) strategies provide a statistically principled way of designing data collection plans that optimally reduce the uncertainty on a given quantity of interest. This thesis focusses on settings where the quantity of interest is a set that is implicitly defined by conditions on some unknown function and one is only able to observe the values of linear operators applied to the function. This setting corresponds to the one encoutered in linear inverse problems and proves to be challenging for SUR techniques. Indeed, SUR relies on having a probabilistic model for the unknown function under consideration, and these models become untractable for moderately sized problem. We start by introducing an implicit representation for covariance matrices of Gaussian processes (GP) to overcome this limitation, and demonstrate how it allows one to perform SUR for excursion set estimation in a real-world 3D gravimetric inversion problem on the Stromboli volcano. In a second time, we focus on extending vanilly SUR to multivariate problems. To that end, we introduce the concept of 'generalized locations', which allows us to rewrite the co-kriging equations in a form-invariant way and to derive semi-analytical formulae for multivariate SUR criteria. Those approaches are demonstrated on a river plume estimation problem. After having extended SUR for inverse problems to large-scale and multivariate settings, we devote our attention to improving the realism of the models by including user-defined trends. We show how this can be done by extending universal kriging to inverse problems and also provide fast k-fold cross-validation formulae. Finally, in order to provide theoretical footing for the developed approaches, show how the conditional law of a GP can be seen as a disintegration of a corresponding Gaussian measure under some suitable condition.