标题： 一种采用拓扑-高斯先验的贝叶斯反潜在问题方法 显示英文标题

标题： A Bayesian approach for inverse potential problem with topological-Gaussian prior

摘要： 本文研究了在椭圆问题中从分布式观测数据重建势系数的问题，并采用贝叶斯框架。 在这种问题中，选择适当的先验分布至关重要，特别是当需要推断的函数具有尖锐间断点时，传统的高斯先验往往不够用。 为了解决这一挑战，我们开发了拓扑先验（TP），这是一种利用持久同调构造的新先验。 所提出的先验使用持久对来表征和记录重构函数的拓扑变化，从而编码关于函数结构和间断点的先验信息。 然而，这种TP先验仅以离散形式存在，这导致在函数空间中不存在明确定义的后验测度。 为了解决这个问题，我们提出了TP-高斯混合先验，在这种先验中，TP部分检测函数中的尖锐间断点，而高斯分布则作为参考测度，确保在函数空间中有明确定义的后验测度。 所提出的TP先验的效果类似于经典的全变差（TV）先验，但由于三个关键优势，它具有更大的灵活性和更广泛的应用范围。 首先，它定义在一个通用的拓扑空间上，使其易于适应更广泛的应用。 其次，持久距离捕获了比离散TV先验更丰富的拓扑信息。 第三，它包含更多可调参数，提供了增强的灵活性，以实现稳健的数值结果。 这些特性使TP先验成为解决涉及具有尖锐间断点的函数的反问题的强大工具。 显示英文摘要

摘要： 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.