This work introduces structure preserving hierarchical decompositions for sampling Gaussian random fields (GRF) within the context of multilevel Bayesian inference in high-dimensional space. Existing scalable hierarchical sampling methods, such as those based on stochastic partial differential equation (SPDE), often reduce the dimensionality of the sample space at the cost of accuracy of inference. Other approaches, such that those based on Karhunen-Lo\`eve (KL) expansions, offer sample space dimensionality reduction but sacrifice GRF representation accuracy and ergodicity of the Markov Chain Monte Carlo (MCMC) sampler, and are computationally expensive for high-dimensional problems. The proposed method integrates the dimensionality reduction capabilities of KL expansions with the scalability of stochastic partial differential equation (SPDE)-based sampling, thereby providing a robust, unified framework for high-dimensional uncertainty quantification (UQ) that is scalable, accurate, preserves ergodicity, and offers dimensionality reduction of the sample space. The hierarchy in our multilevel algorithm is derived from the geometric multigrid hierarchy. By constructing a hierarchical decomposition that maintains the covariance structure across the levels in the hierarchy, the approach enables efficient coarse-to-fine sampling while ensuring that all samples are drawn from the desired distribution. The effectiveness of the proposed method is demonstrated on a benchmark subsurface flow problem, demonstrating its effectiveness in improving computational efficiency and statistical accuracy. Our proposed technique is more efficient, accurate, and displays better convergence properties than existing methods for high-dimensional Bayesian inference problems.

翻译：本文在高维空间的多级贝叶斯推断背景下，提出了用于高斯随机场（GRF）采样的结构保持分层分解方法。现有的可扩展分层采样方法，例如基于随机偏微分方程（SPDE）的方法，通常以降低推断精度为代价来缩减样本空间的维度。其他方法，如基于Karhunen-Loève（KL）展开的方法，虽能实现样本空间降维，但会牺牲GRF表示精度和马尔可夫链蒙特卡洛（MCMC）采样器的遍历性，且在高维问题中计算成本高昂。所提出的方法将KL展开的降维能力与基于随机偏微分方程（SPDE）采样的可扩展性相结合，从而为高维不确定性量化（UQ）提供了一个稳健、统一的框架，该框架具有可扩展性、高精度、保持遍历性并能实现样本空间降维。我们多级算法中的层级结构源自几何多重网格层级。通过构建一种在层级间保持协方差结构的分层分解，该方法实现了从粗粒度到细粒度的高效采样，同时确保所有样本均从目标分布中抽取。在一个基准地下水流问题上验证了所提方法的有效性，证明了其在提升计算效率和统计精度方面的优势。对于高维贝叶斯推断问题，我们提出的技术比现有方法更高效、更精确，并展现出更优的收敛特性。