TY - JOUR
T1 - Emulation of higher-order tensors in manifold Monte Carlo methods for Bayesian Inverse Problems
AU - Lan, Shiwei
AU - Bui-Thanh, Tan
AU - Christie, Mike
AU - Girolami, Mark
N1 - Funding Information:
We thank Patrick R. Conrad for assistance in running the elliptic PDE example. SL is supported by EPSRC Programme Grant, Enabling Quantification of Uncertainty in Inverse Problems (EQUIP), EP/K034154/1 . TB is supported by Department of Energy grants DE-SC0010518 and DE-SC0011118 . MC is partially supported by EPSRC Programme Grant EQUIP, EP/K034154/1 . MG is funded by an EPSRC Established Career Research Fellowship EP/J016934/2 .
Publisher Copyright:
© 2015 Elsevier Inc.
PY - 2016/3/1
Y1 - 2016/3/1
N2 - The Bayesian approach to Inverse Problems relies predominantly on Markov Chain Monte Carlo methods for posterior inference. The typical nonlinear concentration of posterior measure observed in many such Inverse Problems presents severe challenges to existing simulation based inference methods. Motivated by these challenges the exploitation of local geometric information in the form of covariant gradients, metric tensors, Levi-Civita connections, and local geodesic flows have been introduced to more effectively locally explore the configuration space of the posterior measure. However, obtaining such geometric quantities usually requires extensive computational effort and despite their effectiveness affects the applicability of these geometrically-based Monte Carlo methods. In this paper we explore one way to address this issue by the construction of an emulator of the model from which all geometric objects can be obtained in a much more computationally feasible manner. The main concept is to approximate the geometric quantities using a Gaussian Process emulator which is conditioned on a carefully chosen design set of configuration points, which also determines the quality of the emulator. To this end we propose the use of statistical experiment design methods to refine a potentially arbitrarily initialized design online without destroying the convergence of the resulting Markov chain to the desired invariant measure. The practical examples considered in this paper provide a demonstration of the significant improvement possible in terms of computational loading suggesting this is a promising avenue of further development.
AB - The Bayesian approach to Inverse Problems relies predominantly on Markov Chain Monte Carlo methods for posterior inference. The typical nonlinear concentration of posterior measure observed in many such Inverse Problems presents severe challenges to existing simulation based inference methods. Motivated by these challenges the exploitation of local geometric information in the form of covariant gradients, metric tensors, Levi-Civita connections, and local geodesic flows have been introduced to more effectively locally explore the configuration space of the posterior measure. However, obtaining such geometric quantities usually requires extensive computational effort and despite their effectiveness affects the applicability of these geometrically-based Monte Carlo methods. In this paper we explore one way to address this issue by the construction of an emulator of the model from which all geometric objects can be obtained in a much more computationally feasible manner. The main concept is to approximate the geometric quantities using a Gaussian Process emulator which is conditioned on a carefully chosen design set of configuration points, which also determines the quality of the emulator. To this end we propose the use of statistical experiment design methods to refine a potentially arbitrarily initialized design online without destroying the convergence of the resulting Markov chain to the desired invariant measure. The practical examples considered in this paper provide a demonstration of the significant improvement possible in terms of computational loading suggesting this is a promising avenue of further development.
KW - Bayesian Inverse Problems
KW - Gaussian Process emulation
KW - Hamiltonian Monte Carlo
KW - Markov Chain Monte Carlo
KW - Uncertainty quantification
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U2 - 10.1016/j.jcp.2015.12.032
DO - 10.1016/j.jcp.2015.12.032
M3 - Article
AN - SCOPUS:84951760490
SN - 0021-9991
VL - 308
SP - 81
EP - 101
JO - Journal of Computational Physics
JF - Journal of Computational Physics
ER -