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Geometric MCMC for infinite-dimensional inverse problems
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Beskos, Alexandros, Girolami, Mark, Lan, Shiwei, Farrell, Patrick E. and Stuart, A. M. (2017) Geometric MCMC for infinite-dimensional inverse problems. Journal of Computational Physics, 335 . pp. 327-351. doi:10.1016/j.jcp.2016.12.041
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Official URL: http://dx.doi.org/10.1016/j.jcp.2016.12.041
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.
| Item Type: | Journal Article | ||||||||||||||||||||||||
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| Subjects: | Q Science > QA Mathematics | ||||||||||||||||||||||||
| Divisions: | Faculty of Science > Mathematics | ||||||||||||||||||||||||
| Library of Congress Subject Headings (LCSH): | Algorithms, Markov processes, Monte Carlo method, Hilbert space | ||||||||||||||||||||||||
| Journal or Publication Title: | Journal of Computational Physics | ||||||||||||||||||||||||
| Publisher: | Academic Press Inc. Elsevier Science | ||||||||||||||||||||||||
| ISSN: | 0021-9991 | ||||||||||||||||||||||||
| Official Date: | 15 April 2017 | ||||||||||||||||||||||||
| Dates: |
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| Volume: | 335 | ||||||||||||||||||||||||
| Page Range: | pp. 327-351 | ||||||||||||||||||||||||
| DOI: | 10.1016/j.jcp.2016.12.041 | ||||||||||||||||||||||||
| Status: | Peer Reviewed | ||||||||||||||||||||||||
| Publication Status: | Published | ||||||||||||||||||||||||
| Access rights to Published version: | Restricted or Subscription Access | ||||||||||||||||||||||||
| RIOXX Funder/Project Grant: |
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