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 the topological prior (TP), 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 TP 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 TP-Gaussian hybrid prior, where the TP 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 TP prior demonstrates effects similar to the classical total variation (TV) prior but offers greater flexibility and broader applicability due to three key advantages. First, it is defined on a general topological space, making it easily adaptable to a wider range of applications. Second, the persistent distance captures richer topological information compared to the discrete TV prior. Third, it incorporates more adjustable parameters, providing enhanced flexibility to achieve robust numerical results. These features make the TP prior a powerful tool for addressing inverse problems involving functions with sharp discontinuities.

翻译：本文在贝叶斯框架下研究如何从分布式观测数据重构椭圆问题中的势系数。在此类问题中，先验分布的选取至关重要，特别是当待推断函数存在急剧间断时，传统高斯先验往往难以适用。为应对这一挑战，我们提出了一种基于持续同调构建的新型先验——拓扑先验。该先验利用持续对来表征并记录重构过程中的函数拓扑变化，从而编码关于函数结构与间断性的先验信息。然而，拓扑先验仅存在于离散化表述中，这导致函数空间中无法定义良定的后验测度。为解决此问题，我们提出一种拓扑-高斯混合先验，其中拓扑分量用于检测函数中的急剧间断，而高斯分布作为参考测度，确保函数空间中存在良定的后验测度。所提出的拓扑先验展现出与经典全变分先验相似的效果，但凭借三大优势具有更强的灵活性与更广的适用性：其一，该先验定义于一般拓扑空间，易于适配更广泛的应用场景；其二，相较于离散全变分先验，持续距离能捕捉更丰富的拓扑信息；其三，该先验包含更多可调参数，为实现稳健数值结果提供了更强的灵活性。这些特性使得拓扑先验成为处理含急剧间断函数反演问题的有力工具。