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.

翻译：我们在贝叶斯框架下研究从分布式观测数据重构椭圆型偏微分方程中势系数的问题。在此类反问题中，先验分布的选择至关重要，特别是当目标函数存在尖锐间断性时，传统高斯先验难以有效捕捉这些特征。为克服这一局限，我们提出一种基于持续同调（PH）的新型先验，该先验通过持续对量化并编码候选函数的拓扑特征。为确保在无限维空间中定义良好的分布，该先验是相对于高斯参考测度构建的。相较于经典方法的一个重要优势在于：PH先验仅要求未知函数属于合适的拓扑空间，这显著增强了其适用性。数值结果表明，所提出的PH先验优于高斯先验，且相较于经典全变分（TV）先验实现了适度而稳定的改进。