A persistent homology based prior for the potential coefficient reconstruction problem in an elliptic equation

This paper addresses the reconstruction of a potential coefficient in an elliptic problem from distributed observations within the Bayesian framework. In such problems, the selection of an appropriate prior distribution is crucial, particularly when the function to be inferred exhibits sharp discontinuities, as traditional Gaussian priors often prove inadequate. To tackle this challenge, we develop a persistent homology based prior (PH), a new prior constructed using persistent homology. The proposed prior utilizes persistent pairs to characterize and record the topological variations of the functions under reconstruction, thereby encoding prior information about the structure and discontinuities of the function. The PH prior, however, only exists in a discretized formulation, which leads to the absence of a well-defined posterior measure in function spaces. To resolve this issue, we propose a PH-Gaussian hybrid prior, where the PH component detects sharp discontinuities in the function, while the Gaussian distribution acts as a reference measure, ensuring a well-defined posterior measure in the function space. The proposed PH prior exhibits properties analogous to those of the classical TV prior, while providing enhanced flexibility and a wider range of applications. This positions the PH prior as a natural generalization of the TV prior framework.