The Bayesian Finite Element Method in Inverse Problems: a Critical Comparison between Probabilistic Models for Discretization Error

TL;DR

The paper tackles the problem that discretization error in FEM can biasedly distort posterior inference in inverse problems. It evaluates three approaches to model discretization uncertainty: BFEM, which places a Gaussian-process prior on the PDE solution and propagates uncertainty to the posterior in a space orthogonal to the FEM basis; RM-FEM, which randomizes mesh nodes and marginalizes over these perturbations; and statFEM, which learns a model-misspecification term from data. Through two mechanical-test-like experiments (a 1D pullout and a 2D three-point bending problem), BFEM consistently yields more accurate posteriors and avoids overconfidence, outperforming RM-FEM and statFEM, particularly under data-scarce conditions. The results support BFEM as a robust, principled way to propagate discretization uncertainty to Bayesian inverse problems, with practical guidance on mesh strategy and potential for combinations with other error-modeling approaches. The study focuses on linear elasticity and suggests future work extending to nonlinear settings and hybrid error-modeling frameworks.

Abstract

When using the finite element method (FEM) in inverse problems, its discretization error can produce parameter estimates that are inaccurate and overconfident. The Bayesian finite element method (BFEM) provides a probabilistic model for the epistemic uncertainty due to discretization error. In this work, we apply BFEM to various inverse problems, and compare its performance to the random mesh finite element method (RM-FEM) and the statistical finite element method (statFEM), which serve as a frequentist and inference-based counterpart to BFEM. We find that by propagating this uncertainty to the posterior, BFEM can produce more accurate parameter estimates and prevent overconfidence, compared to FEM. Because the BFEM covariance operator is designed to leave uncertainty only in the appropriate space, orthogonal to the FEM basis, BFEM is able to outperform RM-FEM, which does not have such a structure to its covariance. Although inferring the discretization error via a model misspecification component is possible as well, as is done in statFEM, the feasibility of such an approach is contingent on the availability of sufficient data. We find that the BFEM is the most robust way to consistently propagate uncertainty due to discretization error to the posterior of a Bayesian inverse problem.

Figures (11)

• Figure 1: Graphical models of all statistical models considered in this work. A circle is used to indicate a probabilistic variable. If this variable is observed, the circle is shaded. A dot is used to indicate a deterministic (input) variable. Conditional dependencies between variables are indicated by arrows. For more details on probabilistic graph models, the reader is referred to bishop_pattern_2006.
• Figure 1: a) schematic overview of the pullout test. b) FEM solution ${u}^\text{h}{(x)}$ for different mesh sizes $h$. The exact solution given by equation \ref{['eq:pullout-bar-exact-solution']} is shown in black.
• Figure 1: Samples from the FEM posterior distribution $p\lparen{\bm{\theta}}|{\bm{y}}\rparen$ of the three-point bending test inverse problem for six different mesh sizes $h$. The true location of the hole is indicated in black.
• Figure 2: Distributions over the solution field $u{(x)}$ for BFEM and RM-FEM.
• Figure 2: Marginal FEM posterior distributions $p\lparen\theta_i|{\bm{y}}\rparen$ of the three-point bending test inverse problem for six different mesh sizes $h$. The true parameter values are indicated with a black, vertical line.