AI 中文总结
研究基于高斯随机场先验的分层贝叶斯反演中协方差超参数不确定性问题，利用平方指数核的解析解构建卡尔胡宁-勒夫展开，消除超参数更新时重复求解积分特征值问题的计算成本，经优化方法提高精度，应用于达西流模型贝叶斯反演。
AI 中文摘要
具有高斯随机场先验的分层贝叶斯反演解决了协方差超参数中的不确定性，如标准差和相关长度。当高斯随机场由卡尔胡宁-勒夫（KL）展开表示时，基函数通过与协方差核相关的积分特征值问题（IEVP）依赖于这些超参数。超参数更新时IEVP须重复求解，导致分层推理计算成本高。本文聚焦平方指数核，用高斯加权IEVP的解析解构建KL展开，消除超参数更新时IEVP的重复数值解，提高计算效率。虽解析KL展开适用于任意域和维度，但缺乏传统KL展开的均方最优性。为此采用基于优化的方法，选择IEVP中高斯权重函数的标准差以有效降低KL展开的截断误差。一维和二维数值实验表明该选择策略在实际应用中提供了足够精度。此外，解析KL展开允许闭式微分，可通过HMC进行高效后验采样。所提框架应用于稳态达西流模型的贝叶斯反演，成功用弱信息超先验估计了水力传导率场。
英文摘要
Hierarchical Bayesian inversion with Gaussian random field priors addresses uncertainty in covariance hyperparameters, such as the standard deviation and correlation length. When a Gaussian random field is represented by the Karhunen-Loève (KL) expansion, the basis functions depend on these hyperparameters through an integral eigenvalue problem (IEVP) associated with the covariance kernel. Consequently, the IEVP must be solved repeatedly whenever the hyperparameters are updated, leading to significant computational cost in hierarchical inference. In this paper, we focus on the squared exponential kernel and construct the KL expansion using the analytical solution to a Gaussian-weighted IEVP. This analytical KL expansion offers a computationally efficient alternative to the conventional KL expansion by eliminating the repeated numerical solutions of the IEVP during hyperparameter updates. While the analytical KL expansion is applicable to arbitrary domains and dimensions, it does not have the same mean-square optimality as the conventional KL expansion. To address this limitation, we employ an optimization-based approach that selects the standard deviation of the Gaussian weight function in the IEVP to effectively reduce the truncation error of the KL expansion. Numerical experiments in one- and two-dimensional settings show that this selection strategy provides sufficient accuracy for practical applications. Furthermore, the analytical KL expansion admits closed-form differentiation, enabling efficient posterior sampling via HMC. The proposed framework is applied to Bayesian inversion for a steady Darcy flow model, where the hydraulic conductivity field is successfully estimated using weakly informative hyperpriors.
