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

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.