用于解决无限维贝叶斯逆散射问题的持久同调-高斯先验
A persistent-homology-Gaussian prior for solving infinite-dimensional Bayesian inverse scattering problems
AI总结：
研究针对无限维贝叶斯逆散射问题，提出持久同调-高斯（PHG）先验，结合加权持久性正则化项与周期高斯参考测度，经吉布斯倾斜等处理，通过pCN方法采样，能在多噪声条件下准确稳定重建，有效控制拓扑特征。
AI中文摘要：
贝叶斯推理方法已被用于解决函数空间中的逆问题，其中未知参数是无限维的。然而，传统高斯先验在重建实际应用（如障碍物重建）中遇到的不连续或急剧变化的目标函数时仍存在不足。虽然混合先验是一种有前景的解决方案，但在工程应用中开发理论严格且计算易处理的框架仍面临重大挑战。为解决这些问题，我们提出一种持久同调-高斯（PHG）先验，用于在无限维贝叶斯设置中解决声学障碍物散射逆问题，它通过吉布斯倾斜将基于加权持久性的正则化项与周期高斯参考测度相结合。然后，用单位圆上的对数径向函数表示复杂边界，将远场数据重建表述为函数空间逆问题。在Hellinger、总变差和Wasserstein - p度量中建立了所得后验测度的适定性。此外，获得了有限维后验近似的收敛性，并通过预处理的Crank - Nicolson（pCN）方法进行后验采样。数值实验表明，所提出的PHG先验在更广泛的噪声条件下能产生准确且稳定的重建，能明确控制多尺度拓扑特征，与其他传统先验相比性能更好。
英文摘要：
Bayesian inference methods have been developed to address inverse problems in function spaces where the unknown parameters are of infinite dimension. However, conventional Gaussian priors remain inadequate for reconstructing discontinuous or sharply varying target functions encountered in practical applications like obstacle reconstruction. Although hybrid priors have emerged as a promising solution, significant challenges remain in developing theoretically rigorous and computationally tractable frameworks in engineering applications. To address these issues, we propose a persistent-homology-Gaussian (PHG) prior for solving the acoustic obstacle scattering inverse problem in the infinite-dimensional Bayesian setting, which combines a weighted persistence-based regularization term with a periodic Gaussian reference measure through a Gibbs tilt. Then, the complex boundary is represented by a log-radial function on the unit circle, so that the reconstruction from far-field data is formulated as a function-space inverse problem. The well-posedness of the resulting posterior measure is established in the Hellinger, total variation, and Wasserstein-\(p\) metrics. Furthermore, the convergence of finite-dimensional posterior approximations is obtained, and posterior sampling is performed by a preconditioned Crank--Nicolson (pCN) method. Numerical experiments show that the proposed PHG prior yields accurate and stable reconstructions under more extensive noisy conditions, providing explicit control of multiscale topological features and better performance compared to other conventional priors.
