Localization for MCMC: sampling high-dimensional posterior distributions with local structure
Name
1710.07747.pdf
Description
Accepted version
Size
3.12 MB
Format
Adobe PDF
Checksum (MD5)
fa0a7ff2c032696fab3490254746aacd
Author(s)
Morzfeld, M.
Tong, X.T.
Marzouk, Youssef M
Date Issued
January 2019
Journal
Journal of Computational Physics
Publisher
Elsevier BV
Citation
Morzfeld, M., et al. "Localization for MCMC: sampling high-dimensional posterior distributions with local structure." Journal of Computational Physics 380 (March 2019): 1-28 © 2018 Elsevier Inc.
Version
Author's final manuscript
Abstract
We investigate how ideas from covariance localization in numerical weather prediction can be used in Markov chain Monte Carlo (MCMC) sampling of high-dimensional posterior distributions arising in Bayesian inverse problems. To localize an inverse problem is to enforce an anticipated “local” structure by (i) neglecting small off-diagonal elements of the prior precision and covariance matrices; and (ii) restricting the influence of observations to their neighborhood. For linear problems we can specify the conditions under which posterior moments of the localized problem are close to those of the original problem. We explain physical interpretations of our assumptions about local structure and discuss the notion of high dimensionality in local problems, which is different from the usual notion of high dimensionality in function space MCMC. The Gibbs sampler is a natural choice of MCMC algorithm for localized inverse problems and we demonstrate that its convergence rate is independent of dimension for localized linear problems. Nonlinear problems can also be tackled efficiently by localization and, as a simple illustration of these ideas, we present a localized Metropolis-within-Gibbs sampler. Several linear and nonlinear numerical examples illustrate localization in the context of MCMC samplers for inverse problems. Keyword: Markov chain Monte Carlo; Bayesian inverse problems; high dimensions; localization; dimension-independent convergence
MIT Department
Massachusetts Institute of Technology. Department of Aeronautics and Astronautics
Terms of Use
Creative Commons Attribution-NonCommercial-NoDerivs License
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1016/j.jcp.2018.12.008
