Scalable High-Dimensional Bayesian Field Reconstruction with Finite Elements: Application to 3D Porous Media Flow
We present a unified, finite-element-native variational inference framework for very high-dimensional Bayesian spatial field reconstruction in physics-based problems governed by partial differential equations (PDEs) that are nonlinear in the inferred parameters. The framework delivers a full-covariance Gaussian variational posterior, with a probabilistic treatment of all prior and likelihood hyperparameters, on a three-dimensional curved finite-element discretization at a stochastic field dimension exceeding 400000. To our knowledge, this is the first full-covariance variational reconstruction at this scale, complementing the low-rank Hessian-Laplace approaches that dominate extreme-scale Bayesian inversion. The spatial prior is derived from the stochastic PDE (SPDE) connection and formulated natively in terms of finite-element (FE) operators. The sparse Gaussian variational distribution is parameterized via its precision Cholesky factor, with the sparsity pattern inherited from the domain's Laplacian. Unlike covariance-based sparse parameterizations, which encode only short-range correlations, the sparse precision implicitly represents dense posterior covariances through its sparse inverse, yielding smooth, physically plausible samples at O(n) memory cost and enabling direct evidence-lower-bound (ELBO) gradients via the path-derivative (sticking-the-landing) estimator. Natural gradient strategies stabilize convergence, while a variational Bayes expectation-maximization (VB-EM) loop marginalizes all hyperparameters analytically and induces an automatic coarse-to-fine continuation. The framework is demonstrated on Bayesian permeability field reconstruction for a porous-media flow problem, recovering all major spatial features with high fidelity. Algorithmic ablation and comparison with alternative inference methods quantify the improvements over state-of-the-art baselines.