Scalable posterior approximations for large-scale Bayesian inverse problems via likelihood-informed parameter and state reduction
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Willcox_Scalable posterior approximations.pdf
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Author(s)
Cui, Tiangang
Marzouk, Youssef M
Willcox, Karen E
Date Issued
June 2016
Journal
Journal of Computational Physics
Publisher
Elsevier BV
Citation
Cui, Tiangang, et al. “Scalable Posterior Approximations for Large-Scale Bayesian Inverse Problems via Likelihood-Informed Parameter and State Reduction.” Journal of Computational Physics, vol. 315, June 2016, pp. 363–87.
Version
Author's final manuscript
Abstract
Two major bottlenecks to the solution of large-scale Bayesian inverse problems are the scaling of posterior sampling algorithms to high-dimensional parameter spaces and the computational cost of forward model evaluations. Yet incomplete or noisy data, the state variation and parameter dependence of the forward model, and correlations in the prior collectively provide useful structure that can be exploited for dimension reduction in this setting-both in the parameter space of the inverse problem and in the state space of the forward model. To this end, we show how to jointly construct low-dimensional subspaces of the parameter space and the state space in order to accelerate the Bayesian solution of the inverse problem. As a byproduct of state dimension reduction, we also show how to identify low-dimensional subspaces of the data in problems with high-dimensional observations. These subspaces enable approximation of the posterior as a product of two factors: (i) a projection of the posterior onto a low-dimensional parameter subspace, wherein the original likelihood is replaced by an approximation involving a reduced model; and (ii) the marginal prior distribution on the high-dimensional complement of the parameter subspace. We present and compare several strategies for constructing these subspaces using only a limited number of forward and adjoint model simulations. The resulting posterior approximations can rapidly be characterized using standard sampling techniques, e.g., Markov chain Monte Carlo. Two numerical examples demonstrate the accuracy and efficiency of our approach: inversion of an integral equation in atmospheric remote sensing, where the data dimension is very high; and the inference of a h eterogeneous transmissivity field in a groundwater system, which involves a partial differential equation forward model with high dimensional state and parameters.
MIT Department
Massachusetts Institute of Technology. Department of Aeronautics and Astronautics
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Creative Commons Attribution-NonCommercial-NoDerivs License
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DOI of Published Version
https://doi.org/10.1016/J.JCP.2016.03.055
