Bayesian adaptive eigenspace inversion

Abstract We develop a probabilistic analogue of the adaptive eigenspace inversion (AEI) method, an algorithm for the adapting regularisation terms towards the recovery of sharp interfaces, from the perspective of Bayesian inverse problems (BIPs) to enable the quantification of uncertainties in the reconstructions. We show that the adaptations of the regularisation terms in AEI is analogous to adapting the prior distributions in a sequence of BIPs. We present a theoretical result, showing that appropriate convergence of the AEI loop will lead to convergence of the resulting Bayesian posteriors in a Wasserstein distance. Furthermore, we present some numerical experiments to demonstrate the performance and wide applicability of the Bayesian extension of AEI in both linear and nonlinear inverse problems. In particular, we observe in our experiments that the posterior means recover sharp edges in the unknown fields, and the posterior variances are largely concentrated around sharp interfaces, a feature that is also observed in other models for recovery of piecewise constant fields.