This paper studies the formulation, well-posedness, and numerical solution of Bayesian inverse problems on metric graphs, in which the edges represent one-dimensional wires connecting vertices. We focus on the inverse problem of recovering the diffusion coefficient of a (fractional) elliptic equation on a metric graph from noisy measurements of the solution. Well-posedness hinges on both stability of the forward model and an appropriate choice of prior. We establish the stability of elliptic and fractional elliptic forward models using recent regularity theory for differential equations on metric graphs. For the prior, we leverage modern Gaussian Whittle--Matérn process models on metric graphs with sufficiently smooth sample paths. Numerical results demonstrate accurate reconstruction and effective uncertainty quantification.

翻译：本文研究度量图上贝叶斯逆问题的公式化、适定性及数值求解方法，其中图的边表示连接顶点的单向线段。我们聚焦于根据解的有噪观测数据，反演度量图上（分数阶）椭圆型方程的扩散系数这一逆问题。适定性取决于前向模型的稳定性与先验分布的恰当选择。利用度量图上微分方程的正则性理论最新成果，我们建立了椭圆型及分数阶椭圆型前向模型的稳定性。在先验选取方面，我们采用具有充分光滑样本路径的现代高斯Whittle–Matérn过程模型。数值结果验证了该方法可实现精确重构与有效的不确定性量化。