A persistent-homology-based Bayesian prior for potential coefficient reconstruction in an elliptic PDE
We address the reconstruction of a potential coefficient in an elliptic partial differential equation from distributed observations within the Bayesian framework. The choice of prior distribution is crucial in such inverse problems, particularly when the target function exhibits sharp discontinuities that conventional Gaussian priors fail to capture effectively. To overcome this limitation, we introduce a novel prior based on persistent homology (PH), which quantifies and encodes the topological features of candidate functions through their persistent pairs. To ensure a well-defined distribution in infinite-dimensional spaces, the prior is constructed with respect to a Gaussian reference measure. A significant advantage over classical approaches is that the PH prior only requires the unknown functions to belong to a suitable topological space, which substantially enhances its applicability. Numerical results demonstrate that the proposed PH prior outperforms the Gaussian prior and achieves a modest yet consistent improvement over the classical total variation (TV) prior.