TY - JOUR

T1 - Geometric MCMC for infinite-dimensional inverse problems

AU - Beskos, Alexandros

AU - Girolami, Mark

AU - Lan, Shiwei

AU - Farrell, Patrick E.

AU - Stuart, Andrew M.

N1 - Funding Information:
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.
Publisher Copyright:
© 2017 The Authors

PY - 2017/4/15

Y1 - 2017/4/15

N2 - 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.

AB - 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.

KW - Bayesian inverse problems

KW - Infinite dimensions

KW - Local preconditioning

KW - Markov chain Monte Carlo

KW - Uncertainty quantification

UR - http://www.scopus.com/inward/record.url?scp=85012293304&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85012293304&partnerID=8YFLogxK

U2 - 10.1016/j.jcp.2016.12.041

DO - 10.1016/j.jcp.2016.12.041

M3 - Article

AN - SCOPUS:85012293304

VL - 335

SP - 327

EP - 351

JO - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

ER -