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Geometric MCMC for infinite-dimensional inverse problems

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Title: Geometric MCMC for infinite-dimensional inverse problems
Authors: Beskos, A
Girolami, M
Lan, S
Farrell, PE
Stuart, AM
Item Type: Journal Article
Abstract: Bayesian inverse problems often involve sampling posterior distributions on infinite-dimensional function spaces. Traditional Markov chain Monte Carlo (MCMC) algorithms are characterized by deteriorating mixing times upon mesh-refinement, when the finite-dimensional approximations become more accurate. Such methods are typically forced to reduce step-sizes as the discretization gets finer, and thus are expensive as a function of dimension. Recently, a new class of MCMC methods with mesh-independent convergence times has emerged. However, few of them take into account the geometry of the posterior informed by the data. At the same time, recently developed geometric MCMC algorithms have been found to be powerful in exploring complicated distributions that deviate significantly from elliptic Gaussian laws, but are in general computationally intractable for models defined in infinite dimensions. In this work, we combine geometric methods on a finite-dimensional subspace with mesh-independent infinite-dimensional approaches. Our objective is to speed up MCMC mixing times, without significantly increasing the computational cost per step (for instance, in comparison with the vanilla preconditioned Crank–Nicolson (pCN) method). This is achieved by using ideas from geometric MCMC to probe the complex structure of an intrinsic finite-dimensional subspace where most data information concentrates, while retaining robust mixing times as the dimension grows by using pCN-like methods in the complementary subspace. The resulting algorithms are demonstrated in the context of three challenging inverse problems arising in subsurface flow, heat conduction and incompressible flow control. The algorithms exhibit up to two orders of magnitude improvement in sampling efficiency when compared with the pCN method.
Issue Date: 15-Apr-2017
Date of Acceptance: 13-Dec-2016
URI: http://hdl.handle.net/10044/1/66130
DOI: https://dx.doi.org/10.1016/j.jcp.2016.12.041
ISSN: 0021-9991
Publisher: Elsevier
Start Page: 327
End Page: 351
Journal / Book Title: Journal of Computational Physics
Volume: 335
Copyright Statement: © 2017 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
Keywords: Science & Technology
Physical Sciences
Computer Science, Interdisciplinary Applications
Physics, Mathematical
Computer Science
Markov chain Monte Carlo
Local preconditioning
Infinite dimensions
Bayesian inverse problems
Uncertainty quantification
01 Mathematical Sciences
02 Physical Sciences
09 Engineering
Applied Mathematics
Publication Status: Published
Online Publication Date: 2016-12-28
Appears in Collections:Mathematics
Faculty of Natural Sciences