Goal-oriented optimal approximations of Bayesian linear inverse problems
This work provides theoretically optimal approximations for large-scale inverse problems where only a specific quantity of interest is needed, reducing computational cost without sacrificing accuracy.
The paper proposes optimal dimensionality reduction techniques for goal-oriented Bayesian linear inverse problems, focusing on approximating the posterior covariance and mean of a quantity of interest. It proves optimality with respect to geodesic distance, Kullback-Leibler divergence, Hellinger distance, and weighted Bayes risk, and demonstrates effectiveness on a high-dimensional heat transfer problem.
We propose optimal dimensionality reduction techniques for the solution of goal-oriented linear-Gaussian inverse problems, where the quantity of interest (QoI) is a function of the inversion parameters. These approximations are suitable for large-scale applications. In particular, we study the approximation of the posterior covariance of the QoI as a low-rank negative update of its prior covariance, and prove optimality of this update with respect to the natural geodesic distance on the manifold of symmetric positive definite matrices. Assuming exact knowledge of the posterior mean of the QoI, the optimality results extend to optimality in distribution with respect to the Kullback-Leibler divergence and the Hellinger distance between the associated distributions. We also propose approximation of the posterior mean of the QoI as a low-rank linear function of the data, and prove optimality of this approximation with respect to a weighted Bayes risk. Both of these optimal approximations avoid the explicit computation of the full posterior distribution of the parameters and instead focus on directions that are well informed by the data and relevant to the QoI. These directions stem from a balance among all the components of the goal-oriented inverse problem: prior information, forward model, measurement noise, and ultimate goals. We illustrate the theory using a high-dimensional inverse problem in heat transfer.