Title: Dimension-independent likelihood-informed MCMC

Abstract

Many Bayesian inference problems require exploring the posterior distribution of highdimensional parameters that represent the discretization of an underlying function. Our work introduces a family of Markov chain Monte Carlo (MCMC) samplers that can adapt to the particular structure of a posterior distribution over functions. There are two distinct lines of research that intersect in the methods we develop here. First, we introduce a general class of operator-weighted proposal distributions that are well defined on function space, such that the performance of the resulting MCMC samplers is independent of the discretization of the function. Second, by exploiting local Hessian information and any associated lowdimensional structure in the change from prior to posterior distributions, we develop an inhomogeneous discretization scheme for the Langevin stochastic differential equation that yields operator-weighted proposals adapted to the non-Gaussian structure of the posterior. The resulting dimension-independent and likelihood-informed (DILI) MCMC samplers may be useful for a large class of high-dimensional problems where the target probability measure has a density with respect to a Gaussian reference measure. Finally, we use two nonlinear inverse problems in order to demonstrate the efficiency of these DILI samplers: an elliptic PDE coefficient inverse problem and path reconstruction in a conditioned diffusion.

Authors:

Cui, Tiangang ^[1]; Law, Kody J. H. ^[2]; Marzouk, Youssef M. ^[1]

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+ Show Author Affiliations

1. Massachusetts Inst. of Technology (MIT), Cambridge, MA (United States)
2. Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)

Publication Date:

Thu Oct 08 00:00:00 EDT 2015

Research Org.:

Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States)

Sponsoring Org.:

USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)

OSTI Identifier:

1324173

Alternate Identifier(s):

OSTI ID: 1359277

Grant/Contract Number:  

AC05-00OR22725; SC0009297

Resource Type:

Accepted Manuscript

Journal Name:

Journal of Computational Physics

Additional Journal Information:

Journal Volume: 304; Journal Issue: C; Journal ID: ISSN 0021-9991

Publisher:

Elsevier

Country of Publication:

United States

Language:

English

Subject:

97 MATHEMATICS AND COMPUTING; Markov chain Monte Carlo; likelihood-informed subspace; infinite-dimensional inverse problems; langevin SDE; conditioned diffusion