Jakob Scheffels

Jakob Scheffels, Elizabeth Qian, Iason Papaioannou et al.

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, giving rise to a linear inverse problem. 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 and subsequent downstream computations. 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 in O(10^{-10}) in the inverse problem solution with a 10x and 100x lower-dimensional model, respectively.

Han Cheng Lie, Jakob Scheffels, Elisabeth Ullmann

In the design of computational methods for Bayesian inverse problems, costly forward model evaluations make it difficult to sample from or compute the posterior. This motivates the need for approximate forward models that are cheaper to evaluate. We consider reduced-order forward models which exploit the lower-dimensional structure in the Bayesian inverse problem by projecting to the "likelihood-informed subspace" of the parameter space where the prior-to-posterior update is significant. However, the theoretical properties of these reduced-order forward models and their impact on the solution of the Baysian inverse problem are not always well-understood. In this work we consider linear Gaussian inverse problems with a possibly singular prior covariance matrix. We analyse a recently proposed reduced-order model which uses a Petrov-Galerkin projection to likelihood-informed subspaces that arise in optimal low-rank approximations of the posterior covariance matrix. We bound the error in the resulting approximation of the root prior-preconditioned Hessian of the data misfit. Based on this we also bound the errors of the approximate posterior covariance and mean. Our analysis shows that this reduced-order model recovers the exact posterior when the rank of the reduced-order model is equal to the "intrinsic dimension" of the inverse problem, i.e. the rank of the prior-preconditioned Hessian. Two numerical experiments from structural engineering illustrate the performance of our bounds.