Hierarchical Bayesian inversion using the Karhunen-Lo\`eve expansion with analytical eigenpairs of the squared exponential kernel

Abstract

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-Lo\`eve (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.