Title: Scalable posterior approximations for large-scale Bayesian inverse problems via likelihood-informed parameter and state reduction

Abstract

Two prime 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. However, 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 introduce and compare several strategies for constructing these subspaces using only a limited numbermore » 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 heterogeneous transmissivity field in a groundwater system, which involves a partial differential equation forward model with high dimensional state and parameters.« less

Authors:

Cui, Tiangang ^[1]; Marzouk, Youssef ^[1]; Willcox, Karen ^[1]

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1. Massachusetts Inst. of Technology (MIT), Cambridge, MA (United States)

Publication Date:

Tue Mar 29 00:00:00 EDT 2016

Research Org.:

Massachusetts Inst. of Technology (MIT), Cambridge, MA (United States)

Sponsoring Org.:

USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR); Finnish Meteorological Institute

OSTI Identifier:

1548326

Alternate Identifier(s):

OSTI ID: 1325283

Grant/Contract Number:  

SC0009297

Resource Type:

Accepted Manuscript

Journal Name:

Journal of Computational Physics

Additional Journal Information:

Journal Volume: 315; Journal Issue: C; Journal ID: ISSN 0021-9991

Publisher:

Elsevier

Country of Publication:

United States

Language:

English

Subject:

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Inverse problems; Bayesian inference; Dimension reduction; Model reduction; Low-rank approximation; Markov chain Monte Carlo