Samples from general probability distributions are used in many areas of uncertainty quantification.  However, classic approaches for generating such samples, such as Markov chain Monte Carlo (MCMC), can become inefficient when the target distribution has a varying correlation structure (e.g. “banana-shaped” distributions). 

Transport maps have recently entered the Bayesian arena as an alternative to MCMC, but can become computationally intractable when a highly nonlinear map is required to transform the target distribution to a Gaussian distribution.  We propose a hybrid approach that combines MCMC with transport maps.  The resulting technique uses approximate transport maps to capture some non-Gaussian behavior in the target distribution, but ensures exact sampling through a Metropolis-Hastings correction.   After providing some basic background on MCMC and transport maps, we will describe our new map-accelerated MCMC approach, and illustrate its effectiveness on a range of examples.