Preconditioned Crank-Nicolson Markov Chain Monte Carlo Coupled With Parallel Tempering: An Efficient Method for Bayesian Inversion of Multi-Gaussian Log-Hydraulic Conductivity Fields

Geostatistical inversion with quantified uncertainty for nonlinear problems requires techniques for providing conditional realizations of the random field of interest. Many first-order second-moment methods are being developed in this field, yet almost impossible to critically test them against high-accuracy reference solutions in high-dimensional and nonlinear problems. Our goal is to provide a high-accuracy reference solution algorithm. Preconditioned Crank-Nicolson Markov chain Monte Carlo (pCN-MCMC) has been proven to be more efficient in the inversion of multi-Gaussian random fields than traditional MCMC methods; however, it still has to take a long chain to converge to the stationary target distribution. Parallel tempering aims to sample by communicating between multiple parallel Markov chains at different temperatures. In this paper, we develop a new algorithm called pCN-PT. It combines the parallel tempering technique with pCN-MCMC to make the sampling more efficient, and hence converge to a stationary distribution faster. To demonstrate the high-accuracy reference character, we test the accuracy and efficiency of pCN-PT for estimating a multi-Gaussian log-hydraulic conductivity field with a relative high variance in three different problems: (1) in a high-dimensional, linear problem; (2) in a high-dimensional, nonlinear problem and with only few measurements; and (3) in a high-dimensional, nonlinear problem with sufficient measurements. This allows testing against (1) analytical solutions (kriging), (2) rejection sampling, and (3) pCN-MCMC in multiple, independent runs, respectively. The results demonstrate that pCN-PT is an asymptotically exact conditional sampler and is more efficient than pCN-MCMC in geostatistical inversion problems.