Interacting Markov chains algorithms for Bayesian inversion
Résumé
Markov chains Monte
-
Carlo (
MCMC
) methods are popular to
generate
samples of virtually any distribution. They have been successfully
applied in a wide range of problems over the years. However, they
suffer from
slow mixing when the target distribution is high
dimensional and/or
multimodal
. This is often the case in Bayesian
inversion in the field of
geosciences
: the phenomenon under stu
dy
(
resistivity
, pressure, porosity,...) is
generally modeled by a random
field (Gaussian related or not)
discretized
over
a large grid, and the
forward problem may be highly nonlinear.
Recently, the idea of making interact several Markov chains has been
explored.
This approach improves the mixing properties with respect to
classical single
MCMC
. Furthermore, these algorithms can make
efficient use of large CPU
clusters, with a computational cost similar
to standard
MCMC
. In this work,
we expose the princ
iples of
interacting
MCMC
methods and show how to
design algorithms for
Bayesian inversion.
These methods are illustrated on two examples from
geosciences
. The
first is
the history matching problem in reservoir engineering. This
problem terms to
conditio
n a Gaussian random field, describing either
the permeability field or,
when
thresholded
, the
lithofacies
distribution in the reservoir, to fluid flow
data. A preliminary step
consists in
parameterizing
the Gaussian random field.
A low rank
representation
is generally used where the components can be
selected
according to ad
-
hoc
criteria. The second example is an application to first arrival travel time tomography which relies on an ad
-
hoc
parameterization
of the velocity field.