Markov chain Monte Carlo-based approaches for inference in computationally intensive inverse problems
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 = ζ(x) + . Applications include medical and geological tomography, hydrology, and the modeling of physical and biological systems. We consider applications where the physical relationship ζ(x) can be well approximated by detailed simulation code η(x). When the forward simulation code η(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.