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CATEGORIES:Statistics
SUMMARY:Geometric MCMC for infinite-dimensional Bayesian I
nverse Problems - Alexandros Beskos\, University C
ollege London
DTSTART;TZID=Europe/London:20190125T160000
DTEND;TZID=Europe/London:20190125T170000
UID:TALK115909AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/115909
DESCRIPTION:Bayesian inverse problems often involve sampling p
osterior distributions on infinite-dimensional fun
ction spaces. Traditional Markov chain Monte Carlo
(MCMC) algorithms are characterized by deteriorat
ing mixing times upon mesh-refinement\, when the f
inite-dimensional approximations become more accur
ate. Such methods are typically forced to reduce s
tep-sizes as the discretization gets finer\, and t
hus are expensive as a function of dimension. Rece
ntly\, a new class of MCMC methods with mesh-indep
endent convergence times has emerged. However\, fe
w of them take into account the geometry of the po
sterior informed by the data. At the same time\, r
ecently developed geometric MCMC algorithms have b
een found to be powerful in exploring complicated
distributions that deviate significantly from elli
ptic Gaussian laws\, but are in general computatio
nally intractable for models defined in infinite d
imensions. In this work\, we combine geometric met
hods on a finite-dimensional subspace with mesh-in
dependent infinite-dimensional approaches. Our obj
ective is to speed up MCMC mixing times\, without
significantly increasing the computational cost pe
r step (for instance\, in comparison with the vani
lla preconditioned CrankâNicolson (pCN) method). T
his is achieved by using ideas from geometric MCMC
to probe the complex structure of an intrinsic fi
nite-dimensional subspace where most data informat
ion concentrates\, while retaining robust mixing t
imes as the dimension grows by using pCN-like meth
ods in the complementary subspace. The resulting a
lgorithms are demonstrated in the context of three
challenging inverse problems arising in subsurfac
e flow\, heat conduction and incompressible flow c
ontrol. The algorithms exhibit up to two orders of
magnitude improvement in sampling efficiency when
compared with the pCN method.
LOCATION:MR12
CONTACT:Dr Sergio Bacallado
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