Goal-Oriented Optimal Approximations of Bayesian Linear Inverse Problems
Name
1607.01881.pdf
Size
1.27 MB
Format
Adobe PDF
Checksum (MD5)
0fb130a91183fdb0fbb3f235c2dae402
Author(s)
Cui, Tiangang
Tenorio, Luis
Spantini, Alessio
Willcox, Karen E
Marzouk, Youssef M
Date Issued
October 2017
Journal
SIAM Journal on Scientific Computing
Publisher
Society for Industrial & Applied Mathematics (SIAM)
Citation
Spantini, Alessio et al. “Goal-Oriented Optimal Approximations of Bayesian Linear Inverse Problems.” SIAM Journal on Scientific Computing 39, 5 (January 2017): S167–S196 © 2017 Society for Industrial and Applied Mathematics
Version
Author's final manuscript
Abstract
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.
MIT Department
Massachusetts Institute of Technology. Department of Aeronautics and Astronautics
Terms of Use
Creative Commons Attribution-Noncommercial-Share Alike
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1137/16M1082123
