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Geometric MCMC for Infinite-Dimensional Inverse Problems

Beskos, Alexandros and Girolami, Mark and Lan, Shiwei and Farrell, Patrick E. and Stuart, Andrew M. (2017) Geometric MCMC for Infinite-Dimensional Inverse Problems. Journal of Computational Physics, 335 . pp. 327-351. ISSN 0021-9991. doi:10.1016/

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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.

Item Type:Article
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URLURL TypeDescription Paper
Lan, Shiwei0000-0002-9167-3715
Additional Information:© 2017 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license. Received 20 June 2016, Revised 8 December 2016, Accepted 13 December 2016, Available online 28 December 2016. We thank Claudia Schillings for her assistance in the development of adjoint codes for the groundwater flow problem and Umberto Villa for his assistance in the development of adjoint codes for the laminar jet problem. AB is supported by the Leverhulme Trust Prize. MG, SL and AMS are supported by the EPSRC program grant, Enabling Quantification of Uncertainty in Inverse Problems (EQUIP), EP/K034154/1 and the DARPA funded program Enabling Quantification of Uncertainty in Physical Systems (EQUiPS), contract W911NF-15-2-0121. MG is also supported by an EPSRC Established Career Research Fellowship, EP/J016934/2. PEF is supported by EPSRC grants EP/K030930/1 and EP/M019721/1, and a Center of Excellence grant 179578 from the Research Council of Norway to the Center for Biomedical Computing at Simula Research Laboratory. AMS is also supported by an ONR grant N00014-17-1-2079.
Funding AgencyGrant Number
Leverhulme TrustUNSPECIFIED
Engineering and Physical Sciences Research Council (EPSRC)EP/K034154/1
Defense Advanced Research Projects Agency (DARPA)UNSPECIFIED
Army Research Office (ARO)W911NF-15-2-0121
Engineering and Physical Sciences Research Council (EPSRC)EP/J016934/2
Engineering and Physical Sciences Research Council (EPSRC)EP/K030930/1
Engineering and Physical Sciences Research Council (EPSRC)EP/M019721/1
Research Council of Norway179578
Office of Naval Research (ONR)N00014-17-1-2079
Subject Keywords:Markov chain Monte Carlo; Local preconditioning; Infinite dimensions; Bayesian inverse problems; Uncertainty quantification
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Other Numbering System NameOther Numbering System ID
Andrew StuartJ130
Record Number:CaltechAUTHORS:20161220-181119556
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Official Citation:Alexandros Beskos, Mark Girolami, Shiwei Lan, Patrick E. Farrell, Andrew M. Stuart, Geometric MCMC for infinite-dimensional inverse problems, Journal of Computational Physics, Volume 335, 15 April 2017, Pages 327-351, ISSN 0021-9991, (
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:73038
Deposited By: Linda Taddeo
Deposited On:21 Dec 2016 18:41
Last Modified:11 Nov 2021 05:10

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