Likelihood-informed Model Reduction for Bayesian Inference of Static Structural Loads

TL;DR

The paper tackles the computational cost of Bayesian inference for static structural loads by exploiting a likelihood-informed subspace (LIS) to enable projection-based model reduction. It develops a LIS-based projection for linear static systems with unknown right-hand-side forcing, reducing the forward problem to a low-dimensional model whose dimension r scales with the information content of the data. The authors show that LIS-based reductions achieve near-optimal posterior approximations (comparable to the optimal low-rank approach) with significantly lower online costs than full-order models and outperform POD in this context. This enables efficient Bayesian updating in digital twins and health-monitoring scenarios, with practical impact demonstrated on cantilever-bar and tunnel problems where relative errors drop to numerical precision for r equal to the LIS dimension.

Abstract

Bayesian inverse problems use data to update a prior probability distribution on uncertain parameter values to a posterior distribution. Such problems arise in many structural engineering applications, but computational solution of Bayesian inverse problems is often expensive because standard solution approaches require many evaluations of the forward model mapping the parameter value to predicted observations. In many settings, this forward model is expensive because it requires the solution of a high-dimensional discretization of a partial differential equation. However, Bayesian inverse problems often exhibit low-dimensional structure because the available data are primarily informative (relative to the prior) in a low-dimensional subspace, sometimes called the likelihood-informed subspace (LIS). This paper proposes a new projection-based model reduction method for static linear systems that exploits this low-dimensional structure in the setting where the unknown parameter is the right-hand-side forcing. The proposed method projects the governing partial differential equation onto the likelihood-informed subspace, yielding a computationally efficient reduced model that can be used to accelerate the solution of the inverse problem. Numerical experiments on two structural engineering model problems demonstrate that the proposed approach can successfully exploit the intrinsic low-dimensionality of the problem, obtaining relative errors of O(10^{-10}) in the inverse problem solution with a 10x-100x lower-dimensional model.

Figures (9)

• Figure 1: Flowchart diagram of OLR approximation, LIS model reduction and POD model reduction
• Figure 2: Cantilever bar and circle markings depict measurement locations
• Figure 3: Posterior approximation quality for different number of samples considered in the POD basis calculation
• Figure 4: Prior covariance (left), analytical posterior covariance (top center) and the approximation of the posterior covariance obtained using LIS model reduction (top right), OLR dimensionality reduction (bottom center) and POD model reduction (bottom right) for $r=10$
• Figure 5: Quantitative assessment of posterior mean (left) and covariance approximation (right) of LIS model reduction, POD model reduction and OLR dimensionality reduction