资源环境科技发展态势分析平台

Geostatistical inversion with quantified uncertainty for non‐linear 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, non‐linear problem and with only few measurements; and (3) in a high dimensional, non‐linear 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.