Markov chain Monte Carlo-based approaches for inference in
computationally intensive inverse problems
Dave Higdon, Herbie Lee, Chris Holloman
Duke University and Los Alamos National Laboratory
June 2002
A typical setup for many inverse problems is that one wishes to
update beliefs about a spatially dependent set of inputs x
given rather indirect observations y. Here, the inputs and
observed outputs are related by the complex physical relationship y
= zeta(x) + epsilon. Applications include medical and
geological tomography, hydrology, and the modeling of physical
and biological systems. We consider applications where the
physical relationship zeta(x) can be well approximated by
detailed simulation code eta(x).
When the forward simulation code eta(x) is sufficiently fast,
Bayesian inference can, in principle, be carried out via Markov
chain Monte Carlo (MCMC).
Difficulties arise for two main reasons:
-
Even though the code may
accurately represent the physical process, there are a large
number of unknown, but required, inputs that must be calibrated
to match the observed data y.
-
The computational burden of the fastest available forward simulators
is often large enough that approaches for speeding up the MCMC
calculations are required.
This paper develops approaches for specifying effective
low-dimensional representations of the inputs x along with MCMC
approaches for sampling the posterior distribution. In
particular we consider augmenting the basic formulation with
fast, possibly coarsened, formulations to improve MCMC
performance. This approach can be very easily implemented in a
parallel computing environment. We give examples in single
photon emission computed tomography and in hydrology.
Key Words: Multigrid Markov chain Monte Carlo, Metropolis coupled
Markov chain Monte Carlo, spatial statistics, distributed computing
The manuscript is available in postscript and
pdf formats.